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How to find last two digits of a number
by Total Gadha - Saturday, 17 January 2009, 05:33 PM

This brilliant article was contributed to Total Gadha by Ashish Tyagi, a regular TGite. I do not think there can be
easier or more lucid method of finding the last two digits of a number. Kudos to Ashish for this article. We will also
welcome similar contributions from our other users. Do you have some useful or life-saving funda that you would
like to share? Send it to us and we will publish it with your name- Total Gadha




I am dividing this method into four parts and we will discuss each part one by one:
a. Last two digits of numbers which end in one
b. Last two digits of numbers which end in 3, 7 and 9
c. Last two digits of numbers which end in 2
d. Last two digits of numbers which end in 4, 6 and 8
Before we start, let me mention binomial theorem in brief as we will need it for our calculations.
Last two digits of numbers ending in 1
Let's start with an example.
What are the last two digits of 31
786
?
Solution: 31
786
= (30 + 1)
786
=
786
C
0
× 1
786
+
786
C
1
× 1
785
× (30) +
786
C
2
× 1
784
× 30
2
+ ..., Note that all the terms after the
second term will end in two or more zeroes. The first two terms are
786
C
0
× 1
786
and
786
C
1
× 1
785
× (30). Now, the second term will
end with one zero and the tens digit of the second term will be the product of 786 and 3 i.e. 8. Therefore, the last two digits of the second
term will be 80. The last digit of the first term is 1. So the last two digits of 31
786
are 81.
Now, here is the shortcut:

Here are some more examples:
Find the last two digits of 41
2789
In no time at all you can calculate the answer to be 61 (4 × 9 = 36). Therefore, 6 will be the tens digit and one will be the units digit)
Find the last two digits of 71
56747
Last two digits will be 91 (7 × 7 gives 9 and 1 as units digit)
Now try to get the answer to this question within 10 s:
Find the last two digits of 51
456
× 61
567
The last two digits of 51
456
will be 01 and the last two digits of 61
567
will be 21. Therefore, the last two digits of 51
456
× 61
567
will be the
last two digits of 01 ×21 = 21
Last two digits of numbers ending in 3, 7 or 9
Find the last two digits of 19
266
.
19
266
= (19
2
)
133
. Now, 19
2
ends in 61 (19
2
= 361) therefore, we need to find the last two digits of (61)
133
.
Once the number is ending in 1 we can straight away get the last two digits with the help of the previous method. The last two digits are 81
(6 ×3 = 18, so the tens digit will be 8 and last digit will be 1)
Find the last two digits of 33
288
.
33
288
= (33
4
)
72
. Now 33
4
ends in 21 (33
4
= 33
2
×33
2
= 1089 × 1089 = xxxxx21) therefore, we need to find the last two digits of 21
72
. By
the previous method, the last two digits of 21
72
= 41 (tens digit = 2 × 2 = 4, unit digit = 1)
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So here's the rule for finding the last two digits of numbers ending in 3, 7 and 9:
Now try the method with a number ending in 7:
Find the last two digits of 87
474
.
87
474
= 87
472
×87
2
= (87
4
)
118
×87
2
= (69 × 69)
118
× 69 (The last two digits of 87
2
are 69) = 61
118
× 69 = 81 × 69 = 89
If you understood the method then try your hands on these questions:
Find the last two digits of:
1. 27
456
2. 79
83
3. 583
512

Last two digits of numbers ending in 2, 4, 6 or 8
There is only one even two-digit number which always ends in itself (last two digits) - 76 i.e. 76 raised to any power gives the last two digits
as 76. Therefore, our purpose is to get 76 as last two digits for even numbers. We know that 24
2
ends in 76 and 2
10
ends in 24. Also, 24
raised to an even power always ends with 76 and 24 raised to an odd power always ends with 24. Therefore, 24
34
will end in 76 and 24
53
will end in 24.
Let's apply this funda:
Find the last two digits of 2
543
.
2
543
= (2
10
)
54
×2
3
= (24)
54
(24 raised to an even power) ×2
3
= 76 × 8 = 08
(NOTE: Here if you need to multiply 76 with 2
n
, then you can straightaway write the last two digits of 2
n
because when 76 is multiplied with
2
n
the last two digits remain the same as the last two digits of 2
n
. Therefore, the last two digits of 76 × 2
7
will be the last two digits of 2
7
=
28)
Same method we can use for any number which is of the form 2
n
. Here is an example:
Find the last two digits of 64
236
.
64
236
= (2
6
)
236
= 2
1416
= (2
10
)
141
× 2
6
= 24
141
(24 raised to odd power) × 64 = 24 × 64 = 36
Now those numbers which are not in the form of 2n can be broken down into the form 2n ´ odd number. We can find the last two digits of
both the parts separately.
Here are some examples:
Find the last two digits of 62
586
.
62
586
= (2 × 31)
586
= 2
586
× 3
586
= (2
10
)
58
× 2
6
× 31
586
= 76 × 64 × 81 = 84
Find the last two digits of 54
380
.
54
380
= (2 × 3
3
)
380
= 2
380
× 3
1140
= (2
10
)
38
× (3
4
)
285
= 76 × 81
285
= 76 × 01 = 76.
Find the last two digits of 56
283
.
56
283
= (2
3
× 7)
283
= 2
849
× 7
283
= (2
10
)
84
× 2
9
× (7
4
)
70
× 7
3
= 76 × 12 × (01)
70
×43 = 16
Find the last two digits of 78
379
.
78
379
= (2 × 39)
379
= 2
379
× 39
379
= (2
10
)
37
× 2
9
× (39
2
)
189
× 39 = 24 × 12 × 81 × 39 = 92
Now try to find the last two digits of
1. 34
576
2. 28
287
Ashish Tyagi
I am afraid I shall have to end here and leave the rest of it for my CBT Club students. I shall cover some problems based
on this in the CBT Club this week.

If you think this article was useful, help others by sharing it with your friends!
Reply
Re: How to find last two digits of a number
by prateek gupta - Wednesday, 28 January 2009, 06:48 PM
SEEMS AWESOME.......
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Re: How to find last two digits of a number
by rohit bhatnagar - Sunday, 1 February 2009, 10:21 PM
stupendous man !!
gr8 wrk..
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Re: How to find last two digits of a number
by vikas sharma - Wednesday, 18 February 2009, 05:00 PM

hi if problems last digit is 5 like 25
45
den wat will be last digits
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Re: How to find last two digits of a number
by Seshu p - Monday, 23 February 2009, 10:30 PM
Good article.
But, to add a point to above concept,
the last two digits are nothing but the remainder when the number is divided by 100.
Hence, we can use the Remainder Theorem concepts to find last two digits.
Infact, this can be extended to find last three digits(N%1000),
and so on...
**********************************************************
and, the last two digits of any power of a number ending with 5
are 25 or 75.
For eg: 25^45 = (20+5)^5
and, we can apply binomial theorem. Considering only the first two terms (as explained for ending with 1 case), we can find the last two digits
as: 25
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Re: How to find last two digits of a number
by arnab _quant-um - Monday, 2 March 2009, 12:18 AM
gr8.....
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Re: How to find last two digits of a number
by Sam Rox - Monday, 13 April 2009, 01:01 PM

Thanks a lot sir it's awesome.
Just wanted to know for No. ending in 5, the last two digits would be 25 always?
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Re: How to find last two digits of a number
by Sayan Majumder - Monday, 13 April 2009, 05:36 PM
Well Sam,
Let me consider two cases where the numbers are ending with 5.
1. Numbers where the previous digit of 5 is 0 or any even number and
2. Numbers where the previous digit of 5 is any odd number
In 1
st
case if you raise the number to any power you'll always get 25 as the last two digits.
In 2
nd
case if you raise the number to even power you'll get 25, but in case of odd power you'll get 75 as the last two digits.
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Re: How to find last two digits of a number
by suraj sareen - Saturday, 18 April 2009, 01:07 AM
Hi Sir,
I have a doubt i want to find the last 2 digits of 38 to the power 221?
Thanks
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Re: How to find last two di*gits of a number
by sunil goyal - Saturday, 18 April 2009, 07:59 PM
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38^221=38^220 * 38 =(2*19)^220 *38 = 2^220 * 19^220 * 38
(2^10)^22 * (19^4)^55 *38 = 76 * (19^2 * 19 ^2)^55 * 38
76 * (61 * 61)^55 *38 = 76 * (21)^55 *38 = 76 * 01 * 38 =88
so last 2 digit is 88
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Re: How to find last two di*gits of a number
by Sam Rox - Monday, 20 April 2009, 10:43 AM
I have been hearing of the repetition of the last 2 digits after a multiple of 20. i.e the last 2 digits of 2^20 = 2^40, Is this a valid argument??
And what is the sum of 1+4+6+5+11+6+... 200terms??
Thanks
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Re: How to find last two di*gits of a number
by sanjeeb panda - Monday, 20 April 2009, 06:22 PM
Hi can any one explain me how to calculate last 2 digit of 2^2^2008.
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Re: How to find last two digits of a number
by kundan kumar - Tuesday, 21 April 2009, 09:42 AM
2^2^2008?
= 4^2008
=2^4016
=(2^10)^401 * 2^6
=(1024)^401 * 64
we know that 24 raised to odd power ends with 24
therefore 24 * 64
the ans should be. =36
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Re: How to find last two digits of a number
by life goesOn - Wednesday, 22 April 2009, 09:09 AM

2^2^2008 will not translate into 4^2008.
Just to clarify, check the value with 2^3^2
Logically, it should translate into 2^9
But, if we go by your approach, it would translate into 8^2....
IMHO,
2^2^2008, first, lets find the last 2 digit of 2^2008.
last two digit will be 56,
now, we have a number which is 2^xxxx56
it will translate into (2^10)^odd number)*2^6
=(24)^odd number *2^6=24*64=36
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Re: How to find last two digits of a number
by kundan kumar - Wednesday, 22 April 2009, 10:15 PM
thanxs for the correction
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Re: How to find last two digits of a number
by sakshi dadhich - Saturday, 2 May 2009, 07:07 PM
very very nice sir..thanku v much!!!
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Re: How to find last two digits of a number
by Kunal Dongre - Sunday, 10 May 2009, 01:55 PM

Very nice topic Ashish...Thank you very much
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Re: How to find last two digits of a number
by kaushal roonwal - Sunday, 10 May 2009, 08:21 PM
thanks Ashish and TG... its really great method.
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Re: How to find last two digits of a number
by Ashutosh Singh - Thursday, 14 May 2009, 01:00 PM

Thanks Ashi sh for such useful fundas..
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by Gowtham Muthukkumaran Thirunavukkarasu - Friday, 15 May 2009, 04:51 PM

NUMBER_TABLE_FOR_FINDING_LAST_2_DIGITS.doc
NUMBER TABLE FOR FINDING ‘THE LAST 2 DIGITS’
3^4 ends with 81
13^4 ends with 61
23^4 ends with 41
33^4 ends with 21
43^4 ends with 01
53^4 ends with 81
63^4 ends with 61
73^4 ends with 41
83^4 ends with 21
93^4 ends with 01
7^4 ends with 01
17^4 ends with 21
27^4 ends with 41
37^4 ends with 61
47^4 ends with 81
57^4 ends with 01
67^4 ends with 21
77^4 ends with 41
87^4 ends with 61
97^4 ends with 81
9^2 ends with 81
19^2 ends with 61
29^2 ends with 41
39^2 ends with 21
49^2 ends with 01
59^2 ends with 81
69^2 ends with 61
79^2 ends with 41
89^2 ends with 21
99^2 ends with 01
•5^n always end with 25, if n>1
•The product of any 2 numbers ending with 5 always ends with 25
•2^10 ends with 24
•24^n ends with 76, if n is even
24, if n is odd
I hope this will save some time. BY Gowtham Muthukkumaran
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by pankaj jaiswal - Thursday, 21 May 2009, 03:09 PM
nice article...
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by mba 2009 - Monday, 8 June 2009, 12:58 AM
last two digits can also be find out using the cyclicity for ten's place.
digits cyclicity
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2,3,8 20
4,9 10
5 1
6 5
7 4
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Re: How to find last two digits of a number
by sahana v - Thursday, 11 June 2009, 03:41 PM
This is by far the most easily constructed and most elegantly put approach to solving one of the most recurring type of question in CAT.
Thanks to the author for the efforts. Its highly appreciated!
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Re: How to find last two digits of a number
by sahana v - Monday, 15 June 2009, 10:47 AM
Could anybody help with problems of the type
2007^2008^2009 please???
How do we go about this kind of problems?
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Re: How to find last two digits of a number
by Ruchi Gupta - Wednesday, 17 June 2009, 11:08 PM

Hi ,
Thanks for the nice article!!!!
What are the ans for-:
Find the last two digits of:
1. 27
456
2. 79
83
3. 583
512

Is it
1. 89
2. 69
3. 61

Please reply
Thanks

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Re: How to find last two digits of a number
by jinson alex - Wednesday, 17 June 2009, 11:53 PM
Hello Ruchi
The ans will be
1. 61
2. 39
3. 61
Regards
Jinson
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Re: How to find last two digits of a number
by Megha Verma - Thursday, 18 June 2009, 04:51 PM
Being a moron dat im in maths... i feel TG rocks... nothin cud hv been more simpler & enlightning... M lovin every moment of calculation dese
dayz... n if dis is cumin frm me... it means U Guyz r simply awesome...
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Re: How to find last two digits of a number
by iim freak - Sunday, 21 June 2009, 03:28 AM
hi..jinson ..
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i second u wh the below answers::
1. 61
2. 39
3. 61
Best Regards
Randeep
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Re: How to find last two digits of a number
by yogesh bansal - Sunday, 21 June 2009, 11:56 PM

Could anybody help with problems of the type
2007^2008^2009 please???
soln..
cyclicity fr last 2 places fr 7 is 4..
cycle is
07
49
43
01
nd in d power we hv multiple of 4..so, ans will be 01.
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finding last two digits of a number ending wd 5..
by yogesh bansal - Monday, 22 June 2009, 12:01 AM

finding last two digits of a number ending wd 5..
(.......E5)
even
= 25, where E represents even no.
75, otherwise
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Re: How to find last two digits of a number especially for power of 2
by Saumyadeep Das - Monday, 22 June 2009, 01:28 AM
Hi All
Could anyone plz confirm me this discrepancy regarding the statement - "when 76 is multiplied with 2
n
the last two digits remain the same as
the last two digits of 2
n"
I think if you multiply with 2, means 76*2^1 , the last two digits are "52" rather than "02".
Thank you all for the valuable comments.
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Re: How to find last two digits of a number especially for power of 2
by Software Engineer - Monday, 22 June 2009, 11:43 AM
Saumyadeep,
If you need to multiply 76 with 2
n
, then you can straightaway write the last two digits of 2
n
because when 76 is multiplied with 2
n
the last two
digits remain the same as the last two digits of 2
n
. WHERE n is positive integer greater than 1.
- SE
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Re: How to find last two digits of a number
by shiva subramanian - Monday, 22 June 2009, 04:52 PM
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hello sir..i am new to your site and am interested in buying the geometry ebook..what are the topics covered in it sir?is it a comprehensive book
for cat?similar doubts for the numbers e book as well..and regarding the cat cbt club will it include full length mock tests as well?
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Re: How to find last two digits of a number
by ashish gupta - Monday, 22 June 2009, 08:05 PM
kudos to u ashish..
that was brilliant..
thanx a lot...
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Re: How to find last two di*gits of a number
by saurabh yadav - Friday, 26 June 2009, 09:59 AM

NOTE: Here if you need to multiply 76 with 2
n
, then you can straightaway write the last two digits of 2
n
because when 76 is multiplied with 2
n
the last two digits remain the same as the last two digits of 2
n
. Therefore, the last two digits of 76 × 2
7
will be the last two digits of 2
7
= 28)
kindly explain 2^10)^22 * 2^1 = 02[last two digits],but u have performed 2*19=38 ,why????
and therefore [i think ] ans will be 38 not 88.
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Re: How to find last two digits of a number
by sunny jha - Monday, 6 July 2009, 09:29 AM
answer 2 d above questions related 2d problems asked in finding d last 2 digits of nos ending wt 2,4,6,8 r...
1)- 16
2)-12
r dey correct TG sir????
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Re: How to find last two di*gits of a number
by rikant garg - Thursday, 6 August 2009, 02:22 PM
the only problem with 76*2^n rule is when 76*2 gives 152 last two digit is 52 not 02 so we all can note this exception......
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Re: How to find last two digits of a number
by Sanjay J - Monday, 10 August 2009, 07:47 PM

Hi ,
As always with TG articles, a very good article! Thanks Ashish and TG
Can someone explain the following? How to use this ? Please explain with some examples.

last two digits can also be find out using the cyclicity for ten's place.
digits cyclicity
2,3,8 20
4,9 10
5 1
6 5
7 4
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Re: How to find last two digits of a number
by pratyush panda - Wednesday, 12 August 2009, 07:50 PM
thnx for all above concepts...dey will be of great help..
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Re: How to find last two digits of a number
by abhishek tripathi - Friday, 14 August 2009, 11:26 PM
thx ashish 4 dis splendid job.....sir u dnt know how much great work u have done i dnt have any words in mah lexicon 2 describe it simply hats
off 2 u...
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Re: How to find last two digits of a number
by Mallikarjuna Reddy - Thursday, 20 August 2009, 01:19 PM

really useful man.. thnks a lot
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Re: How to find last two digits of a number
by pavithra viswanathan - Saturday, 22 August 2009, 11:49 AM
i didnt get the soln.. can u help me out with it
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Re: How to find last two digits of a number
by Ishika the best - Saturday, 22 August 2009, 09:50 PM
the most lucid concept i hv evr studied..thanks tg sir
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Re: How to find last two digits of a number
by PAVAN C - Saturday, 22 August 2009, 10:18 PM
thank you for this great article
Find the last two digits of:
1. 27
456
2. 79
83
3. 583
512
i am getting
1. 61
2. 39
3. 61
as answer, plz explain the steps if i am wrong.
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Re: How to find last two digits of a number
by Kunal Parekh - Sunday, 6 September 2009, 11:26 PM

Dear Ashish,
Thank you very much for this valuable post.
Regards.
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Re: How to find last two digits of a number
by manoj kumar - Monday, 7 September 2009, 12:02 AM
Thanks a lot TG...really a very helpful article...nd thanks to ashish as well..
Regards
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Re: How to find last two digits of a number
by Sandeep Ghosh - Thursday, 10 September 2009, 02:59 AM
Great concepts
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Re: How to find last two digits of a number
by ravish sehgal - Friday, 11 September 2009, 12:55 PM
awesome article ...i havnt seen calc easier than this one ....nice ....
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Re: finding last two digits of a number ending wd 5..
by anurag saxena - Friday, 11 September 2009, 04:58 PM
dis method z nt valid..plz check it agn...mc
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Re: How to find last two digits of a number
by Pallav Jain - Friday, 11 September 2009, 11:27 PM
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Hello

These methods are just awsome. time saving, fast and verbal calculation.
Great Job Done.
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Re: How to find last two digits of a number
by Pallav Jain - Friday, 11 September 2009, 11:29 PM
Dude i m getting the same answers. ok tell me 1 thng hw much time u took to solve these 3 ques.?
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Re: How to find last two digits of a number
by Pallav Jain - Friday, 11 September 2009, 11:52 PM

Hello Sahana
I have got 01 as the answer of this Problem. Is this Right ?
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Re: How to find last two digits of a number
by tapas mani shyam - Saturday, 12 September 2009, 01:32 AM
hey pallav ..
can you please explain how you arrived at the answer??
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Re: How to find last two digits of a number
by Ashish Sharma - Saturday, 12 September 2009, 03:29 PM
awefome
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Re: How to find last two digits of a number
by Pallav Jain - Sunday, 13 September 2009, 11:23 AM

What if we cud have 2003^2004^2005 and so on....
and need to find last 2 digits.....
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Re: How to find last two digits of a number
by parasar datta - Sunday, 13 September 2009, 04:35 PM
the last 2 digits of 2003^2004^2005 are 81
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Re: How to find last two digits of a number
by Pallav Jain - Monday, 14 September 2009, 08:40 PM

Hi can anyone please tell me the detailed Solution for this Question.
2003^2004^2005.....
LAST 2 DIGITS
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Re: How to find last two digits of a number
by Netra Mehta - Monday, 14 September 2009, 10:49 PM
Hi Pallav!!
the Ans m gettin for 2003^2004^2005..... is 81.
Plz tell me d correct ans...
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Re: How to find last two digits of a number
by Netra Mehta - Monday, 14 September 2009, 11:01 PM
See Pallav,, it goes like that...
firstly consider 2005^2006^2007.....
Here the last 2 digits ll cum out to b 05...Since 5^(any no.) is alwyz 25
Now consider (2004)^xxxx25 it can b written as
(4*501)^xxxx25
= 4^xxxx25 * (501)^xxxx25
=2^xxxx50 * 01
= (2^10)^xxx5 * 01
=(1024)^xxxx5 * 01
= 24 * 01 = 24
Finally the question is reduced to 2003^xxx24
which is of the form
= (03 * 03 * 03 * 03)^ xxx1 or 03 * 03 * 03 * 03)^ xxx6
= (81)^ xxx1 or (81)^xxx6
In both of the cases last 2 digits cum out to b 81
Hence ans. is 81
I hope my approach is rite n u understood dat Pallav!!
Still if m wrong do tell me...
Netra
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Re: How to find last two digits of a number
by Pallav Jain - Tuesday, 15 September 2009, 08:40 PM

Hi Netra,
Answer is Abstly Correct. But Dont you think This Method is Quite Long..
1st Step is Fine abt 5 power 25 all tht. 2nd we can do (2000+4) power.............25. 2000 wl gv 00 so we js need 2 check 4 ( as u know).
2 power.........50. will gv 24 as last digit as u explained.
den 2000+3) power 24. We js need to consider 24 as we can get Lst 2 digits wid tht. 3 power 24. OR 81 power 6.
so it will gv 81 As the Last 2 digits.
Am i Right?
Pallav
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Re: How to find last two digits of a number
by Pallav Jain - Tuesday, 15 September 2009, 08:42 PM

Hello Guys,
Is there any one interested in Material Swaping. Who think that he or she have complete their Material so that we can Swap our Material for
Some Time..
Reply.....
Pallav
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Re: How to find last two digits of a number
by Sujit Patra - Wednesday, 16 September 2009, 06:33 AM
nice one
indeed very useful....
thanx
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Re: How to find last two digits of a number
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by Software Engineer - Wednesday, 16 September 2009, 12:39 PM

2003^2004^2005
L(100)=20
2004^2005 mod 20
p=4 and q=5
2004^2005 mod 5
(-1)^2005 mod 5
-1 mod 5
R=4x=5y-1
x=1, R=4
2003^4 mod 100
81 mod 100
- SE
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Re: How to find last two digits of a number
by Ashish Tripathi - Wednesday, 16 September 2009, 03:04 PM

Hi Pallav,
I think we all have gone too far with it, here's what i did....
For 2003^2004^2005 expression we need to calculate the last two digits, hence the last two digits for it will be the same as the last two digits
for 03^04^05 or 3^4^5 for which the last two digits come up as 81.
Regards
Ashish
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Re: How to find last two digits of a number
by ROHIT K - Wednesday, 16 September 2009, 03:08 PM

i second d above solution..that is the right way to do..
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Re: How to find last two digits of a number
by Saurabh Kumar - Wednesday, 16 September 2009, 04:40 PM

Ashish,

I cannnot think of a simpler approach then you have suggested. I don't have the words to say thanks. Your contribution will be remembered by
all CAT aspirants whjo read this article
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Re: How to find last two digits of a number
by Software Engineer - Wednesday, 16 September 2009, 05:08 PM

I think that will fail in case of 2222^4000^6666.
Last two digit of each term,
22^00^66
= 22^0
= 1
but, 2222^4000^6666 = 76 mod 100.
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Re: How to find last two digits of a number
by ROHIT K - Wednesday, 16 September 2009, 05:37 PM
@ S.E
That is very true..But, I think, we should change approach for specific kind of questions to arrive @ solution in a much quicker way.
The normal method is always there for all problems.
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by Raju Singh - Thursday, 17 September 2009, 08:59 PM
Hi guys/Gowtham Muthukkumaran Thirunavukkarasu
In this forum one of the reply posted by Gowtham Muthukkumaran Thirunavukkarasu Posted on 15 May 2009.
He (u) said that product of any numbers ending with 5 always ends in 25.
I think it is not necessary as it could be either 25 or 75. Take for example:
65 x 35 = 2275
85 x 35 = 2975 or many more ...
which in not 25 but 75.
Tell me if 'am wrong or not getting what he wants to say.
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by ROHIT K - Thursday, 17 September 2009, 09:15 PM
Hi Raju,
Good observation..
He made a mistake..
Anyways, the product of any two numbers ending with 05 always ends in 25
Rohit
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by Raju Singh - Thursday, 17 September 2009, 11:45 PM
ya Rohit
Indeed more general way (Product of two numbers ending in 5)is:
E5 x E5 = 25 as last two digits, where E is even numbers (0,2,4,6,8) preceding 5
O5 x O5 = 25 as last two digits, where O is odd numbers (1,3,5,7,9) preceding 5
E5 x O5 = 75 as last two digits.
Guys could u tell me what is cyclicity method. I don't knw this method. Would it be easier method or not?
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Re: How to find last two digits of a number
by Netra Mehta - Friday, 18 September 2009, 02:46 AM
Yup Pallav!
u r right!!
Dat wz indeed a short method...
Thnx alot
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Re: How to find last two digits of a number
by akshita sharma - Sunday, 27 September 2009, 04:57 PM
can u explain it in detail!!!!
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Re: How to find last two digits of a number
by Aastha Chalana - Monday, 28 September 2009, 12:11 AM

Hi SE,
can u pls elaborate how to solve this question (last two digits of 2003^2004^2005) with the help of euler's theorem. i am getting 43 as the
answer by that method..small pointers of the solution i did
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phi(100) = 40 then solving further, i get Remainder[(2003^40k +5)/100]
which gives 43
pls elaborate. also i have read whole of the ur article on the remainders.. and religiously solved each n every question by my own and too using
just eulers theorem, and i ahve some more doubts will post in that article only. thanks!!
waiting for the reply of the query asked above.
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Re: How to find last two digits of a number
by Amandeep Singh - Tuesday, 29 September 2009, 03:42 PM
Hi SE,
Can you elaborate what is L(100) = 20 and
2004^2005 mod 20
p=4 and q=5
2004^2005 mod 5
(-1)^2005 mod 5
-1 mod 5
R=4x=5y-1
x=1, R=4
these steps. What is the method you are using?
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Re: How to find last two digits of a number
by Pallav Jain - Tuesday, 29 September 2009, 06:18 PM
Hi Guys,
can anyone help me wid the detailed solution of this question.
6^83 + 8^83 DIVIDED BY 49
WHAT IS THE REMAINDER ?
Pallav
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Re: How to find last two digits of a number
by ROHIT K - Wednesday, 30 September 2009, 11:12 AM
Hi pallav,
I had already replied to this problem in the Remainder post.
I think, I had replied to your post itself.
Please check in that thread.
Hope this helps..
Rohit
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Re: How to find last two digits of a number
by Pankaj Kumar - Wednesday, 30 September 2009, 12:10 PM
hi, how number 2^xxxx56 will translate into (2^10)^odd number)*2^6 ??
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Re: How to find last two digits of a number
by Pallav Jain - Wednesday, 30 September 2009, 12:21 PM
Hi Rohit
Thanx for your reply but i cud'nt get your explanation. Can you please explain me in Detail.
Thanks
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Re: How to find last two digits of a number
by ROHIT K - Wednesday, 30 September 2009, 07:36 PM
Hi Pallav,
You did not understand either of the two methods I explained in that post??
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I wonder if you have checked the right post which I wrote just after you asked me to explain in detail.
anyways I am pasting the post again..if you did not understand the below post, I think it is better that TG or some other TGite from the forum
explain you. I can't give a more detailed explanation than this..sorry..if you have any problem understanding any step you can reply to that
post in the Remainders thread.
by ROHIT K
-
Wednesday,
16
September
2009, 11:08
AM

Sorry..I was lazy and ate steps .
I will try to explain using two methods:
1. Binomial Theorem
2. Euler Theorem
1. Binomial Theorem Method (I think everyone knows this one)
(6^83+8^83)%49= ((7-1)^83 + (7+1)^83)%49
Expand (a-b)^n and (a+b)^n
You will notice that all the terms would be divisible by 49 except last 2,
since all the terms would have power of 7 more than 2 except the last 2 which would be 7^1 and 7^0 (not divisible by 49).
So we r left with 83C82 * 7 - 1 + 83C82 * 7 + 1(last two terms of both expansion)
The question now boils down to (2*83*7) % 49
Divide by 7 ==> Question becomes (2*83)%7 viz. 166%7=5
Since we divided by 7, we have to multiply the intermediate remainder by 7.
Therefore,
The Final Remainder = 5 * 7 = 35
2. Euler Method
Calculating Euler Totient of 49 written as phi(49).
phi(49)=49*(1-(1/7))=6*7=42
Therefore we have,
(6^42k)%49=1 and (8^42k)%49=1 (By Euler Theorem) k is any natural number
Now,
Question is (6^83+8^83)%49
Adjusting powers to suit the above method.
((6^84 * 6^(-1)) + (8^84 * 8^(-1))) %49
6^84%49 =1 And 8^84 %49=1 (By Euler Theorem)
Hence the question becomes:
((6^-1) + (8^-1))%49
=(8*48^-1 + 6*48^-1)mod49 (by taking L.C.M and solving)
=48^-1(8+6)mod49
=48^-1*14mod49
=-1^-1*14mod49 ..................(48%49=-1)
=14*-1mod49
=-14mod49
=35............(49-14=35)
Hence the Final Remainder=35
Hope this explanation is clear.
Rohit
p.s. It would be better if this topic is discussed in the relevant thread..Hope you agree..
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Re: How to find last two digits of a number
by avinash nishchhal - Thursday, 1 October 2009, 04:58 PM
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its great......
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Re: How to find last two digits of a number
by kanika kumar - Sunday, 4 October 2009, 05:27 PM
amazing ! simply amazingggggggggggggg
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Re: How to find last two digits of a number
by NAHUSH PHADKE - Tuesday, 6 October 2009, 11:55 AM
si mpl y awesome and amazi ng!!!!!!
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Re: How to find last two digits of a number
by manish singh - Thursday, 8 October 2009, 12:58 PM

ultimate!!
cheers!!
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Re: How to find last two digits of a number
by Software Engineer - Thursday, 8 October 2009, 03:45 PM

Aastha, rather than applying Euler's Theorem, use the methods described in Remainders.
Amandeep, please refer this.
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Re: How to find last two digits of a number
by Aastha Chalana - Sunday, 11 October 2009, 03:03 AM

Hi SE,
thanks for replying as i have been waiting for it for long now! i was trying to avoid carmichael's thm and my effort was to solve all remainder
question by EUlers thm.
but as u suggested i studied the Carmichaels's thm,,, as i found it easier.. actually i observed,, it reduces the totient further, lesser that euler
totient and as we reduce the totient, it leaves lower value of when divided by the power hence earier calc in finding remainder of a number
lamda(16) is 4 and phi(16) is 8...
hope i made sense.. thanks again...
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Re: How to find last two digits of a number
by jai singh - Sunday, 11 October 2009, 10:33 AM

awesome.....
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Re: How to find last two digits of a number
by Tony Boruah - Tuesday, 10 November 2009, 01:29 AM

Great article...thank u!!!!
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Re: How to find last two digits of a number
by archana Agarwal - Thursday, 12 November 2009, 06:09 PM
gr8 article thankzzz
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Re: How to find last two digits of a number
by amar goswami - Saturday, 14 November 2009, 10:46 PM

@life goes on .. awesome awesome awesome.. wat i was doing was very lengthy..
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Re: How to find last two digits of a number
by shweta k - Sunday, 15 November 2009, 12:34 AM

GR8 wrk.... i admire u 4 such wondrful work....... kudos 2
u.... keep it up....
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Re: How to find last two digits of a number
by amar goswami - Sunday, 15 November 2009, 10:11 PM
Dear CAT Aspirants,
I would advise you all to practice these short cuts thoroughly before you go to take on CAT. I read this topic and remainders topic
yesterday. Today i gave CL mock test and i was so happy to see a question on last two digits of a number. But trust me, i could not remember
what was the short cut for solving the question. Wasted 2 minutes on this but all i could remember was '24' , 'euler theorm' and 'lambeda' .
Switched back to basics and problem got solved. Thank GOD fetched me 3 marks..
So in nutshell ..practice and practice more.. ATB
Below is problem which came in test:
find the 10
th
digit of the number(i.e. 2nd from right) :
6
11
7
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Re: How to find last two digits of a number
by smruty panigrahi - Monday, 16 November 2009, 10:36 AM

I also went through same. Became desperate. Wasted 5-6 min and ultimately bowed out.
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Re: How to find last two digits of a number
by amar goswami - Monday, 16 November 2009, 11:13 AM
Dont worry Smruty. Its good if we do all our mistakes in mocks and take a lesson from each of them. Ultimately what matters is our
performance in CAT paper. As per your reply it seems you had left the question, i would recommend you to go through remainders post. It
would give you better insight to solve such questions. .
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Re: How to find last two digits of a number
by deepika gandhi - Monday, 16 November 2009, 01:37 PM

Find the last two digits of:
1. 27
456
2. 79
83
3. 583
512
i am getting
1. 61
2. 39
3. 61
can u pls xplain me how did u get 2nd n 3rd answers 39 n 61...i am getting 79 n 41....
pls reply soon
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Re: How to find last two digits of a number
by amar goswami - Monday, 16 November 2009, 02:27 PM
Hi Deepika,
Although the question was not asked to me but i hope u wont mind if i
answer this. It goes like this:
79
83
= (79
2
)
41
* 79 = xx41
41
*79
Now you need to refer to the rule given above to find the last two
digits of a number ending with 1.
From this rule :
4*1 =4 = 10th digit and unit digit =1
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hence our expression becomes
41*79 = xx39 which gives 39 as last two digits
Now try to solve the 3rd question following
the same rule and m sure u will crack it
Rgds
Amar
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Re: How to find last two digits of a number
by deepika gandhi - Monday, 16 November 2009, 02:52 PM
hey thnx amar.....
i got d answer...i did d last step rong...
thnx 1 again....
n total gadha,,,,u rock man...........
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Re: How to find last two digits of a number
by deepika gandhi - Monday, 16 November 2009, 03:14 PM

What if we cud have 2003^2004^2005 and so on....
and need to find last 2 digits.....
ans: firstly consider 2005^2006^2007.....
Here the last 2 digits ll cum out to b 05...Since 5^(any no.) is alwyz 25
Now consider (2004)^xxxx25 it can b written as
(4*501)^xxxx25
= 4^xxxx25 * (501)^xxxx25
=2^xxxx50 * 01
= (2^10)^xxx5 * 01
=(1024)^xxxx5 * 01
= 24 * 01 = 24
Finally the question is reduced to 2003^xxx24
which is of the form
= (03 * 03 * 03 * 03)^ xxx1 or 03 * 03 * 03 * 03)^ xxx6
= (81)^ xxx1 or (81)^xxx6
In both of the cases last 2 digits cum out to b 81.....
can u pls simplify it...i dint get it
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Re: How to find last two digits of a number
by amar goswami - Monday, 16 November 2009, 03:40 PM

it means that only last two digits of 2003 will contribute to last two digit of the final product. Hence 2003
xxx24
can be written as 03
xxx24
which in turn can be written as (03
4
)
xxx1
or (03
4
)
xxx6
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because any number xxxx24 when divided by 4 will give 1 or 6 as unit digit of its quotient.
Hope this helps.
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Re: How to find last two digits of a number
by Pallav Jain - Tuesday, 17 November 2009, 04:40 AM
Hi Amar
I am getting 56 as the last 2 digit of your mock question.
6^11^7
Am I Right?
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Re: How to find last two digits of a number
by amar goswami - Tuesday, 17 November 2009, 05:10 PM

Absolutely correct Pallav
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Re: How to find last two digits of a number
by Pallav Jain - Tuesday, 17 November 2009, 07:05 PM
Hi Amar
you are enrolled in which center of CL. & how was the Mock 8 level this time. I am also a CL student but I didnt take Mock 8.
Cheers !!!!!
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Re: How to find last two digits of a number
by amar goswami - Tuesday, 17 November 2009, 09:27 PM
I am enrolled with Noida centre, test series student. Proc mock 8 was easy as compared to previous tests. 22 questions in each section with +3
/-1 marking scheme. DI was the easiest, QA average and EU on the tougher side(coz of tricky questions).
Hope we are not violating terms and conditions of use of Totalgadha.com by discussing a competitor's (CL) stuff here. However if i did, i am
very sorry and i simply dint mean to do so. TG is the best and everyone knows this, one who doesn't has not visited the site properly.
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Re: How to find last two digits of a number
by Anand Omar - Tuesday, 17 November 2009, 10:01 PM

My Dear
Last 2 digit for any power of 5 will always 25.

Try it...
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Re: How to find last two digits of a number
by amar goswami - Wednesday, 18 November 2009, 03:49 PM
Anand
I dont know who is 'dear' here but i can give u an example of what ppl are talking :
5^3 = 125 --- i
15^3 = 3375 --- (ii)
(ii) implies that if 10th digit is odd(1 here) and power(3 here) is odd then last two digits are not 25.
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Re: How to find last two digits of a number
by rohit dwivedi - Friday, 20 November 2009, 05:54 PM
Hi Deepika and Netra,
In regard of question 2003^2004^2005
I am not able to get the first line of both solutions.
That is we have to find tens and unit digits of 2004^xxxx25 .
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From where this 25 had come when it was initially 2005
Thanks In Advance ..
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Re: How to find last two digits of a number
by shankar gorkhe - Friday, 20 November 2009, 06:52 PM

uhhh!!
Last 2 digit for any power of 5
5^1 =05
5^2 =25
5^9 =1953125
5^11 =48828125
5^19 =.......

not multiple of 5.....
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Re: How to find last two digits of a number
by rahul gupta - Friday, 20 November 2009, 07:34 PM

Bang on the target!!!!!!!!!!
Must say Fundooooooooo article TG.......
U rock!!!!!
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Re: How to find last two digits of a number
by versha jhunjhunwala - Tuesday, 24 November 2009, 12:14 PM

hey.......please tell me how did you get that....
last digit 56.....
p lease elaborate it......
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Re: How to find last two digits of a number
by Pallav Jain - Tuesday, 24 November 2009, 10:47 PM

versha jhunjhunwala
6^11^7
(2^11^7)*(3^11^7)
(2^11^7)
ok as per TG's Concept..
11^7 last 2 digits wud be ----------71
nw 2^-------71
which can be written as (2^10)^7 * 2^1
1024^7*2^1 = 24*2 as the last 2 digits.... 48
same can be done with 3
3^11^7
3^---------71
(3^4)^17 * 3^3
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81^17 * 27
61*27
last 2 digits are 47
nw multiply 48*47
So we get 56 as the last 2 digits of the ques.
Hope its understandable
Cheers !!!!!
Pallav Jain
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Re: How to find last two digits of a number
by versha jhunjhunwala - Wednesday, 25 November 2009, 11:45 AM

Thanks ...Pallav....
I understood it ....
but is it necessary to frst calculate lke u did ...11^7 n den fnd it 6^that power ...

I mean if i went lke dat ..
[{6^11}^7]
so ...6^11 n cyclicity of 6 in case of ten's digit is 5 ..so
[{6^5*2 }*6^1]^7
[___56]^7
n we will get ...
___16
so d ans wud be ...16 (is it wrong ...?)

2. 2003^2004^2005
in dat case:
[{2003^4}^501]^2005
[{___81}^501]^2005
[__81]^2005
41 wud be d ans...
But ans is 81 ...so my approach is wrong or wat??



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Re: How to find last two digits of a number
by versha jhunjhunwala - Wednesday, 25 November 2009, 12:48 PM
Sry ...the ans shud be .._01 according to me...
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Re: How to find last two digits of a number
by Pallav Jain - Thursday, 26 November 2009, 02:31 PM

i m really sorry varsha i didnt get ur explanation.
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Re: How to find last two digits of a number
by kanika kalra - Friday, 27 November 2009, 04:00 AM

hii ol
What will be the last digit of
?
can u solve dis 4 me!!
kanika
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Re: How to find last two digits of a number
by tapas mani shyam - Friday, 27 November 2009, 10:12 AM
Hi Kanika
Is the answer 64??
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Re: How to find last two digits of a number
by Pallav Jain - Friday, 27 November 2009, 11:50 AM
hii ol
What will be the last digit of ?
can u solve dis 4 me!!
kanika
Answer is 6
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Re: How to find last two digits of a number
by tapas mani shyam - Friday, 27 November 2009, 10:54 PM

i did it again..again gt ans as 4..
can u solve it here pallav??
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Re: How to find last two digits of a number
by kanika kalra - Saturday, 28 November 2009, 05:39 AM
answer s 4
evn i got 6 !! datz y m askin plz teme tapas plzz
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Re: How to find last two digits of a number
by tapas mani shyam - Saturday, 28 November 2009, 11:58 AM
Pardon me cz I hav neva posted anywhere..so dunno how to make special mathematical symbols..hehehe..but i hope this is understandable...
start from the first term:
in the first term 4
5
= 1024. next 3
1024
= [81]
256
. Now last two digits of this number (with the method given by TG sir) = 81
So, the first term reduces to 2
---81
.
Writing it as [[2]
10
]
--8
* 2 = [24]
even power
* 2 = ----76 * 2 = ----52
Now take the second term. Start from the top:
3
5
= 243. Next we have 15
243
= 3
243
* 5
243
= [81]
60
* 3
3
* 5
243

Now this reduces to (----01) * 27 * (----25)
{ as 05 raised to any power always ends in 25}
this again reduces to (----27) * (-----25) = (-----75)
Now the final 3
(-----75)
= [81]
(---18)
* 3
3
= (-----41) * (27) = (-----07)
So, finally multiplying the two terms (---52) * (---07) we have the units digit as 4 and tens digit as 6
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Re: How to find last two digits of a number
by kanika kalra - Saturday, 28 November 2009, 04:24 PM
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thanku thanku thanku
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Re: How to find last two digits of a number
by arun ilangovan - Saturday, 28 November 2009, 11:24 PM

hi...it can be solved using the reminder theory also
R(2^3^4^5/10)...
let's move from the bottom...
1) we need 3^4^5 in the form 4K+reminder, (since 2^4k divided by 10 gives reminder 1)
2) to write 3^4^5 in the form 4K+reminder, we need to write 4^5 in the form 2k+reminder (since 3^2k divided by 4 gives reminder 1)
3) but 4^5 is always of the form 2k
4) so, 3^4^5 becomes 3^2k..
5) when 3^2k is divided by 4 (as required by step 2)the reminder is 1...so 3^2k can be written as 4K+1
6) so 2^3^4^5 can be written as 2^(4K+1) and when divided by 10, the reminder is 2 (because 2^4k will give reminder 1)...so the last digit of
2^3^4^5 is 2...
similarly the last digit of the next term is 7..
so the product is 14 and hence the last digit is 4!!!
this method can be used to find the last digit of a number containing any number of powers...
hope, i didn't confuse anyone
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Re: How to find last two digits of a number
by kanika kalra - Sunday, 29 November 2009, 03:38 AM

bt frst method s far betr
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Re: How to find last two digits of a number
by Raju Singh - Sunday, 29 November 2009, 12:17 PM
@ Kanika n Tapas,
use this shortcut method in the second case of 3^15^3^5.
as 3^5 = 243
therefore, 15^243= xx75 (as last two digits). How? see
If number ends in 5 then use this format
(X5)^n for n>1
1) if X is even then for all value of n the last two digits always be 25.
2) if X is odd then two case:
a) if n is even then also last two digits will be 25.
b) if n is odd then last two digits will be 75.
This is alway applicable if number ends in 5
So in case 15^243 as X is odd and so is n hence last two digit is 75.
There I have seen number of people say that (XX5)^n always ends in 25! no some time it may be 75. If single digit 5^n then it always end in
25
I think kanika had done same mistake thats why get 6 as answer.
hope it helps u someway.
Thanks
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Re: How to find last two digits of a number
by Rahul Bhat - Wednesday, 10 March 2010, 09:13 PM
Hey TG...Simply fantastic...
My ans-
1. 27^456 = 61
2. 79^83 = 39
3. 583^512 = 61
4. 34^576 = 16
5. 28^287 = 12
Rahul.
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Re: How to find last two digits of a number
by vinay mudgil - Wednesday, 21 April 2010, 06:12 PM
Awesome article Mr. Ashish [though I know i am little late in admiring this]....
Thank you
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by dwaipayan Biswas - Sunday, 25 April 2010, 10:17 AM
2^1= 02
2^20=1048576
Therefore
2^21= 2097152
therefore the cycle for 2 cannot be 20
For 3 & 8 it's 20. because 3^20= .......01 & 8^20= ........76
for 6 also I find the same thing
6^1=06
6^5=7776
6^6=46656
So the cycle is not repeating after 6^5

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Re: How to find last two digits of a number
by pallavi srivastava - Thursday, 3 June 2010, 10:40 PM
hi if problems last digit is 5 like 25^45 then d last two digit answer will be 25
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by pallavi srivastava - Thursday, 3 June 2010, 10:50 PM

awesome.. wow how did u find it its truly helpful.... n saves alot of time too
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Re: How to find last two digits of a number
by TANUJ GUPTA - Friday, 16 July 2010, 09:56 PM
superb....
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Re: How to find last two digits of a number
by chetan chandrashekar - Thursday, 19 August 2010, 06:11 PM
3^4^5 last two digits is 01.
3^4 is 81
81^5 last two digits is 01
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Re: How to find last two digits of a number
by Rahul Sinha - Sunday, 22 August 2010, 08:36 PM

Hi All,
Could you please let me know how to solve below questions?

1.) (12^55/3^11)+(8^48/16^18) will give the digit at unit place as
a) 4
b) 6
c) 8
d) 0
Ans - (a)
2.) [x] denotes the greatest integer value just below x and {x} its
fractional value.The sum of [x]^2 and {x}^1 is 25.16.Find x
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a)5.16
b)-4.84
c)Both (a) and (b)
d)4.84
Ans - (c)
3.) The power of 45 that will exactly divide 123! is
a)28
b)30
c)31
d)59
e)29
Ans - (c) ---- But here i am getting answer (a)
4.) Three numbers are such that the second is as much as lesser than the third
as the first is lessser than the second.If the product of the two smaller
number is 85 and the product of two larger number is 115 find the middle
number.
a)9
b)8
c)12
d)30
e)10
Ans - (e)
Thanks
Rahul
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Re: How to find last two digits of a number
by TG Team - Monday, 23 August 2010, 01:12 PM

Hi Rahul
1) (12
55
/3
11
) + (8
48
/16
18
) = (3
44
)(4
55
) + 2
72
= (a1)(b4) + c6 = d0. (d)
2) [x]
2
+ {x} = 25.16 = 25 + 0.16 = (5)
2
+ 0.16 = (-5)
2
+ 0.16
So x = [x] + {x} = 5 + 0.16 = 5.16 or -5 + 0.16 = 4.84. (c)
3) 45 = 3
2
*5 and 123! contains 3
41+13+4+1
= 3
59
or 9
29
and 5
24+4
= 5
28
so 45
28
. (a)
4) Let the three numbers be a, b, c and given is: c - b = b - a also ab = 85 and bc = 115.
So ab + bc = b(a + c) = b(2b) = 2b
2
= 85 + 115 = 200 = 2*10
2
. So b = 10 (e)
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Re: How to find last two digits of a number
by Rahul Sinha - Monday, 23 August 2010, 03:57 PM

Hi kamal,
Thank you for your quick response.It seems easy now the way you explained it.
For q. 1) I also got answer as (d) but book (Arun sharma's) says it as (a)
For q. 3) I also got answer as 28 but book says 31.
Rest are correct and I got it.

Thank you once again

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Re: How to find last two digits of a number
by manish gupta - Thursday, 26 August 2010, 12:55 AM
for 2nd ques 4.84 is not possible because [x]=-5 and {x}=48
then [x]^2+{x}=25+.48=25.48
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option a is correct.
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Re: How to find last two digits of a number
by manish gupta - Thursday, 26 August 2010, 01:23 AM
ans is 4 .....
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Re: How to find last two digits of a number
by swayam aahuja - Saturday, 25 September 2010, 11:03 PM
ans will be 01.
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by nishant lal - Thursday, 2 December 2010, 12:35 PM

Hi Gowtham Muthukkumaran,
In your post you said "The product of any 2 numbers ending with 5 always ends with 25" but see this:
15x25=375
25*25=625
25*35=875
25*45=1125
... and so on
I think it should be " The product of any 2 numbers ending with 5 always ends with 25
if sum of their digits in 10s place is even and it ends with 75 if sum of their
digits in 10s place is odd." this is for 2 digit numbers.
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by P BHASKAR - Thursday, 9 December 2010, 12:04 PM
Find the last two digits of:
2. 79^83
answer i am getting is
79^2^41*79
(79*79=6241)
so 41^41*79
then 4*1 =4
answer is 49
i don know i am correct or wrong please reply tg
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by TG Team - Thursday, 9 December 2010, 01:31 PM

Hi Bhaskar
79
83
= (79)(79)
82
≡ (79)(21)
82
mod100 ≡ (79)(41) mod100 ≡ 39 mod100.
Kamal Lohia
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by P BHASKAR - Saturday, 11 December 2010, 11:29 PM
thanks tg team
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How to find last two digits of a number
by Saumya Satpathy - Saturday, 15 January 2011, 03:24 PM

hii everyone..
I am new to this site and its awesome for we CAT aspirants. Intially I was using the remainder theorem method to find out the last two digits.
But after knowing these methods/shortcuts..its further reducing the time and effort to calculate last two digits.
what i think using both therse methods simultaneously one can solve these type of questions in a very short time.
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Re: How to find last two digits of a number
by Bikash Sahu - Wednesday, 19 January 2011, 11:35 AM

Finding the last two digit makes sense when one understands the concept underneath n the number pattern it follows....
It can give some edge over No. Pattern as the CAT itself lies in NUMBERS only.
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Re: How to find last two digits of a number
by HARSHAL KULKARNI - Sunday, 15 May 2011, 08:55 PM

Awesome article as well as Comments. Both helped me a lot.
Cheers
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Re: How to find last two digits of a number
by rajiv nakirikanti - Wednesday, 18 May 2011, 02:51 PM

superbbbb

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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by swagata mukherjee - Thursday, 19 May 2011, 09:21 PM
To TG Sir,
I am from Mumbai intending to join the CAT CBT club & have sent a cheque to that effect through speed post 3 days ago.I have no idea
whether it reached TG or not.How do I know?
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by T Riddler - Friday, 20 May 2011, 05:52 PM
please mail the following details at [email protected]
1. cheque no
2. correspondence address at which you sent the post.
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by swagata mukherjee - Tuesday, 24 May 2011, 06:57 PM
Sir i mailed as u asked me to.Still no clue whether the cheque reached u or not.
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Re: finding last two digits of a number ending wd 5..
by Narayanan Santharaman - Wednesday, 6 July 2011, 05:03 PM
How it works yar.....
(....O5)^Odd = 75
(...E5)^Odd/Even =25
(...O5)^Even = 25
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by somit srivastava - Sunday, 18 September 2011, 12:24 AM
Just take the sum of tens place.. even gives 25 odd 75.. more generalized..
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by Niraj Gohil - Wednesday, 5 October 2011, 12:11 AM
hi everyone..............
how to find last two digits of multiplication 1089 * 1089??
any shortcuts........
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Re: How to find last two digits of a number
by lakshmi k - Thursday, 6 October 2011, 09:35 PM

Let a = 1! + 2! + 3! + . . . . .. . .. 99! + 100! Number of digits in the number ‘a’ are
a.137
b.283
c.314
d.189
e.none of these
How to do this problem?
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by lakshmi k - Thursday, 6 October 2011, 09:37 PM

Multiply the last 2 digits of the numbers ie, 89*89 = 7921 so last 2 digits of1089*1089 is 21.

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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by TG Team - Friday, 7 October 2011, 11:46 AM

Hi Lakshmi
It becomes easier if you understand that finding last two digits of a number is same as finding remainder of the number
with 100. So in this case we have:
1089×1089 ≡ (89×89) mod100 ≡ (-11)(-11) mod100 ≡ 21 mod100.
So required last two digits are 21.
Kamal Lohia
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by Manish Rana - Friday, 14 October 2011, 12:31 PM
why 26^n always ends with 76 if n>1....sir please reply me on [email protected]
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by TG Team - Friday, 14 October 2011, 02:05 PM

Hi manish
Look carefully. It says 24ⁿ ends in 76 as last two digits if n is even otherwise it ends in 24 only. (Here n is a natural
number).
Reason is that 76 is a special number (2-digit automorphic number), which when raised to any natural number power
ends in same last two digits i.e. 76 always. (76ⁿ ≡ 76 mod100).
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Also, there is one more speciality with 76 as follows: 76 × 2ⁿ ≡ 2ⁿ mod100 (iff n > 1).
Kamal Lohia

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Re: finding last two digits of a number ending wd 5..
by sandhya bharti - Wednesday, 26 October 2011, 10:19 PM
hi yogesh...
its always 25..... n nt 75.
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by amit jain - Thursday, 14 June 2012, 05:43 PM
Let a = 1! + 2! + 3! + . . . . .. . .. 99! + 100! Number of digits in the number ‘a’ are
a.137
b.283
c.314
d.189
e.none of these
How to do this problem?
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Re: How to find last two digits of a number
by bittu vishal - Wednesday, 20 June 2012, 01:37 AM
Last digit of any number ending with 5 will be 5 itself
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Re: How to find last two digits of a number
by ashutosh kumar - Wednesday, 20 June 2012, 06:23 PM

just awesome...article
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Re: How to find last two di*gits of a number
by Anindya Sudhir - Saturday, 14 July 2012, 10:26 AM

1+4+6+5+11+6+... 200terms
=>(1+6+11+16+...100terms)+(4+5+6+7+...100terms)
=30200
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Re: How to find last two digits of a number- NUMBER TABLE- Gowtham
by Praveen Patel - Monday, 16 July 2012, 03:41 PM

Hi ,this is really helpful.
thanks Gowtham.
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Re: How to find last two digits of a number
by neha aggarwal - Wednesday, 18 July 2012, 09:33 PM
Hi Sir,
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How to find last two digits of 14^14^14??
Thanks,
neha
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Re: How to find last two digits of a number
by TG Team - Thursday, 19 July 2012, 10:39 AM

Hi Neha
Finding last two digits of a number is same as finding its remainder with 100 i.e. 4 and 25. Now given number is divisible
by 4, so we just need to find its remainder with 25 and then combine.
We know that 14 and 25 are co-prime, so 14
20
= 1 mod 25.
Also 14
14
= 20k + 16
=> 14
14^14
=14
16
mod25 = 4
8
mod25 = 2
16
mod25 = -(2
6
) mod25 = -64 mod25 = 11 mod25 = 36 mod25.
As number is divisible by 4, its last two digits are 36.
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Re: How to find last two digits of a number
by neha aggarwal - Thursday, 19 July 2012, 08:22 PM
Hi Sir,
Thanks for the solution..
Just want to ask though 11 is positive remainder with 25 u hv added 25 to it since obviously for 14 number should end with 6???
And another question:
Whats the last two digit for:
11^11!+12^12!+13^13!+.... + 80^80!
Regards,
Neha
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Re: How to find last two digits of a number
by vinayak m - Saturday, 4 August 2012, 06:34 PM
@neha
as in 11^11!+12^12!+.........+80^80!
11! = 11*10*9*.........*1
it will form 10*5*4*2=400 means power of each term will b multiples of hundred
so 11^11! will end in no.of zeroes.
hence last two digits of above sum wil be zero
so Ans.= 00
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Re: How to find last two digits of a number
by Rhythm Goyal - Tuesday, 7 August 2012, 06:28 PM

For: last two digits of 14^14^14??
Where am i going wrong?
-> 14^14^14 = 14^(2*7)^14
consider (2*7)^14 => 2^14 * 7^14
=> 2^10 * 2^4 * (7^4)^3 * 7^2
=> __24 * 16 * __01^3 * 49
=> __16
Therefore 14^14^14 => 14^__16
=> 2^__16 * 7^__16
=> (2^10)^odd no * 2^6 * (7^4)^4k
=> __76 * 64 * (__01)^4k
=> 64

why am i getting 64? Is it correct?
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Re: How to find last two digits of a number
by TG Team - Tuesday, 7 August 2012, 09:19 PM

Hi Rhythm Goyal
Everything was fine except second-last step where it should be 24 in place of 76 and the answer will be 36.
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Re: How to find last two digits of a number
by Rhythm Goyal - Wednesday, 8 August 2012, 06:06 PM

Oh... Thats so stupid of me... I have verified my calcluations thrice before posting. Still got same wrong answer all the 3 times.
Thanks a lot Sir. I may now not comitt such in CAT.
Sir, I need to discuss regarding Regular batch / CBT. Where can we discuss?
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Re: How to find last two digits of a number
by Mohd Arafat - Wednesday, 19 September 2012, 03:57 PM
Any power of five (>1) always has 25 as last 2 digits
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Re: How to find last two digits of a number
by shyam DASH - Wednesday, 19 September 2012, 09:21 PM
(2957^3661)=????
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Re: How to find last two digits of a number
by siddharth saha - Thursday, 4 October 2012, 08:24 PM
(2957^3661)=----57
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Re: How to find last two digits of a number
by sandip shaw - Saturday, 6 October 2012, 10:22 PM
awesome work by sir but sad to see that more knowledge given to premium students.
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Re: How to find last two digits of a number
by Abhinav Bhardwaj - Sunday, 7 October 2012, 03:35 PM
Hi,
I used this approach.
the no is 9483^67483
(83^4)1687*83^3
Now,
(83^4)^1687 results in 41 as the last two digits.
83^3 results in 87
we multiply 87 and 41 and we get 67..?Bt dats incorrect.
Plz explain??
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Re: How to find last two di*gits of a number
by rahul runthala - Tuesday, 16 October 2012, 04:14 PM

i calculated this way : (38)
221
=(19*2)
221
=19
221
*2
221
=(19)*(19
220
)*(2)*(2
220
)
=19*(19
2
)
110
* 2(2
10
)
22
=19*(361)
110
*2(24)
22

=19*(last digit is 61)
110
*2( last digit is 76)
=19*(61)
110
*2(76)
=19*(01)*2(76)
now if we multiply:19*01*02*76=last is comming 88 Ok, but
as u have mentioned 2
n
*76= last two no. of 2
n
,therefore 19*01*2
1
=last digit is 44. how ...
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Re: How to find last two digits of a number especially for power of 2
by rahul runthala - Tuesday, 16 October 2012, 04:31 PM
ya i too faced the plb ....when cal. =19*01*2*76,it should 19*02, but the ans is not comming
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