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When 51^25 is divided by 13, the remainder obtained is:

A. 12 B. 10 C. 2 D. 1 E. 0

\(51^{25}=(52-1)^{25}\), now if we expand this expression all terms but the last one will have \(52=13*4\) in them, thus will leave no remainder upon division by 13, the last term will be \((-1)^{25}=-1\). Thus the question becomes: what is the remainder upon division -1 by 13? The answer to this question is 12: \(-1=13*(-1)+12\).

Hello Experts, I am not able to understand why the answer is 1 in the first case (65+1)/13 where as it is 12 in the second case (65-1)/13. Is it because of minus (-)? If it is so, then the result should be (9-1)*4 in the second example in the link provided by you.

Please help me understand this concept. Thanks

The remainder in the first case is 1 whereas the remainder in the second case is -1. Why? Because the last term of the binomial will be (-1)^25 in the second case. I strongly suggest that you check out this post to understand binomial: http://www.veritasprep.com/blog/2011/05 ... ek-in-you/

How do we handle a negative remainder such as -1 since the options will only give us positive remainders? That is also explained in detail in the post but let me add a small explanation here as well:

When I divide 32 by 10, I get a remainder of 2 - you know that. Can I also say that the remainder can also be said to be -8 (if negative remainders were allowed)? It's something like this: Say I have $32 and 10 people in front of me. I can give each person $3 and I will be left with $2 (or I can say that my balance is +2). Or I can give each person $4 and I will be poorer by $8 (i.e. my balance will be -8. I would have given $8 from my own pocket). So when I divide a number by 10, I can say that my remainder could be 2 or it could be -8.
_________________

When 51^25 is divided by 13, the remainder obtained is:

A. 12 B. 10 C. 2 D. 1 E. 0

\(51^{25}=(52-1)^{25}\), now if we expand this expression all terms but the last one will have \(52=13*4\) in them, thus will leave no remainder upon division by 13, the last term will be \((-1)^{25}=-1\). Thus the question becomes: what is the remainder upon division -1 by 13? The answer to this question is 12: \(-1=13*(-1)+12\).

Answer: A.

Hey Bunuel,

I am trying to understand this concept what happens if

66^25 is divided by 13 - then how will the above method work?

[65 + 1]^25 then will remainder be 1?

Check out the link I have given above. It discusses the use of Binomial Theorem to solve such questions. And yes, 66^25 is (65+1)^25 and since 65 is divisible by 13, the remainder will be 1 in this case.

On the other hand, if you have 64^25, then you get (65 - 1)^25 so here, remainder will be (-1) i.e. 12
_________________

Quite amazing as the question above entirely contradicts GMAC's and several other prep site's and material's assurance that for GMAT you need not have a real awesome Maths background.

I even never heard of Binomial theorem in my entire life and gave 2 hours to understand it. Nothing clear for me. Can anyone elaborate the complex method of the specific theorem OR some other way to solve the problem.

Here is the thing - GMAC and all test prep companies are absolutely correct. You don't need an awesome Math background. Also, such questions can be easily solved by figuring out the pattern - not this one since calculating powers of 51 (51^2, 51^3 ...) etc is quite painful - but such question with smaller numbers can be easily solved without using Binomial theorem.

Then why do such questions appear in GMAT material? Because either some test prep companies go overboard in making tougher questions or these questions are picked from non-GMAT prep material.

Then why are we discussing Binomial theorem here? Because it is a great way to solve all such questions (either small numbers or large) and it is not difficult to understand. If you understand what \((a + b)^2 = a^2 + b^2 + 2ab\) means, you are halfway there already! That is what I have tried to do in this post: http://www.veritasprep.com/blog/2011/05 ... ek-in-you/ I have tried to explain Binomial in very layman terms and just enough to help you solve all such questions in under a minute.
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Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink]

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15 Mar 2013, 18:27

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Hello Experts, I am not able to understand why the answer is 1 in the first case (65+1)/13 where as it is 12 in the second case (65-1)/13. Is it because of minus (-)? If it is so, then the result should be (9-1)*4 in the second example in the link provided by you.

Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink]

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23 May 2013, 22:23

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Quite amazing as the question above entirely contradicts GMAC's and several other prep site's and material's assurance that for GMAT you need not have a real awesome Maths background.

I even never heard of Binomial theorem in my entire life and gave 2 hours to understand it. Nothing clear for me. Can anyone elaborate the complex method of the specific theorem OR some other way to solve the problem.

Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink]

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23 May 2013, 23:20

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genuinebot85 wrote:

Quite amazing as the question above entirely contradicts GMAC's and several other prep site's and material's assurance that for GMAT you need not have a real awesome Maths background.

I even never heard of Binomial theorem in my entire life and gave 2 hours to understand it. Nothing clear for me. Can anyone elaborate the complex method of the specific theorem OR some other way to solve the problem.

Bunuel and Karishma have already shown the best method. However, you can do this problem in another way too.

The remainder when 51 is divided by 13 can be either (-1) or 12. Assuming that you are not comfortable with negative remainders, we can then restate the problem as : Remainder of\(\frac{(51^{25})}{13}\)= Remainder of\(\frac{(12^{25})}{13}\). Now notice that \(12^2\) = 144 and Remainder of\(\frac{144}{13}\)= 1. Thus, Remainder of\(\frac{(12^{25})}{13}\)= \([(144)^{12}*12]/13\) = Remainder of \(\frac{12}{13}\)= 12.

Assuming that you are comfortable with negative remainders, Remainder of \(\frac{(51^{25})}{13}\)= Remainder of\(\frac{(-1)^{25}}{13}\)= Remainder of -1/13 = 12.
_________________

Quite amazing as the question above entirely contradicts GMAC's and several other prep site's and material's assurance that for GMAT you need not have a real awesome Maths background.

I even never heard of Binomial theorem in my entire life and gave 2 hours to understand it. Nothing clear for me. Can anyone elaborate the complex method of the specific theorem OR some other way to solve the problem.

Bunuel and Karishma have already shown the best method. However, you can do this problem in another way too.

The remainder when 51 is divided by 13 can be either (-1) or 12. Assuming that you are not comfortable with negative remainders, we can then restate the problem as : Remainder of\(\frac{(51^{25})}{13}\)= Remainder of\(\frac{(12^{25})}{13}\). Now notice that \(12^2\) = 144 and Remainder of\(\frac{144}{13}\)= 1. Thus, Remainder of\(\frac{(12^{25})}{13}\)= \([(144)^{12}*12]/13\) = Remainder of \(\frac{12}{13}\)= 12.

Assuming that you are comfortable with negative remainders, Remainder of \(\frac{(51^{25})}{13}\)= Remainder of\(\frac{(-1)^{25}}{13}\)= Remainder of -1/13 = 12.

Actually, you are using Binomial too.

How do you explain Remainder of\(\frac{(51^{25})}{13}\)= Remainder of\(\frac{(12^{25})}{13}\)?

Binomial leads to this equality!
_________________

How do you explain Remainder of\(\frac{(51^{25})}{13}\)= Remainder of\(\frac{(12^{25})}{13}\)?

Binomial leads to this equality!

I don't think I am using Binomial. Maybe you are correct.

However, I would try to prove the above point, not using Binomial. Let me know if I went wrong somewhere.

--> \(51^{25} = 13Q_1 + R_1\)

and \(12^{25} = 13Q_2 + R_2\)

with all the understood notations.

I want to prove that \(R_1\) = \(R_2\)

I will assume they are equal and replace the value of \(R_1\) in the first equation by that of \(R_2\)

Thus, \(51^{25} = 13Q_1 +12^{25} - 13Q_2\)

or \(51^{25}-12^{25} = 13(Q_1-Q_2)\)

Thus, if I could show that \(51^{25}-12^{25}\) is divisible by 13, my assumption would be correct.

Now,\(x^n-a^n\), is always divisible by (x-a), x and a are integers, n is odd.

Thus,\(51^{25}-12^{25}\) is always divisible by (51-12) = 39 --> 13*3. Thus, it is divisible by 13.

In the spirit of a healthy discussion, I would like to point out that you can certainly prove that the two are equal since they ARE equal. It can be done in many ways. The point is that when you see \(51^{25}\), what makes you think of \(12^{25}\)? You think of it because 51 = 39 + 12. You separate out the part which is divisible by 13 and take control of the rest. Why?

Because \((39 + 12)^{25}\), when divided by 13 will have the same remainder as \(12^{25}\) because every term in this expansion is divisible by 39 except the last term which is \(12^{25}\). You understand this intuitively and that is all binomial is about.
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Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink]

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26 May 2013, 03:30

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This is one of the best questions and best replies..really liked it..Although i took around an hour to understand binomial theorem in Karishma's blog
_________________

“Confidence comes not from always being right but from not fearing to be wrong.”

So,we're left out with 4 and remainder for 2^83 divided by 9 is 5.

P.S: can you please clarify that how 8^28 = 9m + 1 (where m is some positive integer) ?

Yes, 5 is correct. Good job. You get \(4*(9 - 1)^{27}\) which gives a remainder of \(4(-1)^{27} = -4\) When divisor is 9, a remainder of -4 is equivalent to a remainder of 5.

"P.S: can you please clarify that how 8^28 = 9m + 1 (where m is some positive integer) "

\(8^{28} = ( 9 - 1)^{28} = [9 + (-1)]^{28}\) When you expand it, you get \(9^{28} + 28*9^{27}*(-1) + ....... (-1)^{28} = 9^{28} - 28*9^{27} + ... + 1\) All terms in the expansion will have 9 as a factor except the last term (which is 1). In the expansion, some terms will be positive and some will be negative. Notice that overall the sum must be positive because the sum is equal to \(8^{28}\) which is positive. when you add/subtract multiples of 9, you will get a multiple of 9 as the answer.

So expansion becomes 9m + 1 (which is equal to \(8^{28}\) so m must be positive)
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Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink]

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24 Jul 2013, 09:47

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Bunuel wrote:

LM wrote:

When 51^25 is divided by 13, the remainder obtained is:

A. 12 B. 10 C. 2 D. 1 E. 0

\(51^{25}=(52-1)^{25}\), now if we expand this expression all terms but the last one will have \(52=13*4\) in them, thus will leave no remainder upon division by 13, the last term will be \((-1)^{25}=-1\). Thus the question becomes: what is the remainder upon division -1 by 13? The answer to this question is 12: \(-1=13*(-1)+12\).

Answer: A.

Hello Bunuel,

I understood the solution to this Question quite well. But I need your inputs on the following. Can this Question be done by first finding the cyclicity of 51^25 which will be 1^ 25 and when 1^ 25 is divided 13 should give remainder 1 and not 12. What am I missing here ??

Consider another Q2 question 2^86/9 ? Cyclicity of 2 is 4 so dividing the power of 2 i.e 86/4 we get remainder 4 {cyclicity 2,4,8,6}

While in this Question we can get remainder straight away

Similarly another Question Q3 " What is remainder when 66^25/13 ?" Discussed in this thread

Now again cyclicity of 6 is 1 and hence remainder should be 6. However the answer seems to be 1 to this question.

Take another Q4 " what is remainder when 2^83 is divided by 9" taken from Karishma's blog on veritas prep Again Cyclicity of 2 is 4 so 2^ 83 will end up in 8 and hence remainder should be 8

Alternatively using other method discussed in Quant Forum 2^83-----> 2^3*2^80------> 2^3*4^40------> 2^3*8^20

Consider 8^20 can written as (9-1)^20 so remainder will be (-1)^20/9 = 1 and multiplying this by 8 we get remainder 8.Is this solution correct ??

If so can you elaborate that why in Q2 & Q4 we are getting the right answer but not in Q1 (original Q) and Q3?

Many Thanks!!!

Mridul
_________________

“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”

When 51^25 is divided by 13, the remainder obtained is:

A. 12 B. 10 C. 2 D. 1 E. 0

\(51^{25}=(52-1)^{25}\), now if we expand this expression all terms but the last one will have \(52=13*4\) in them, thus will leave no remainder upon division by 13, the last term will be \((-1)^{25}=-1\). Thus the question becomes: what is the remainder upon division -1 by 13? The answer to this question is 12: \(-1=13*(-1)+12\).

Answer: A.

Hello Bunuel,

I understood the solution to this Question quite well. But I need your inputs on the following. Can this Question be done by first finding the cyclicity of 51^25 which will be 1^ 25 and when 1^ 25 is divided 13 should give remainder 1 and not 12. What am I missing here ??

Please elaborate on red part. What does this mean? How are you going to do that? ...
_________________

Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink]

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19 May 2014, 09:37

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While solving this question, thought process should be like this 2^3 = 8 which can be written as (9-1) Since power of 2 i.e. 83 is not a multiple of 3 we have to write 2^83 as (2^2)(2^81) because 81 is divisible by 3. Now , 2^81 can be written as (2^3)^27 or (8)^27 or (9-1)^27. In the expansion of (9-1)^27 all the terms will be divisible by 9, except last term i.e. (-1)^27 = -1 Since the second last term will be 9, if we combined last two term, we will get 9-1=8 . This 8 will be remainder as all other terms will have a multiple of 9. We can write the number as 9k+8. Multiply this with 2^2 i.e. 4. we will get 36k+32 or 36k + 27 +5 First two terms are divisible by 9 and this will give 5 as a remainder. Hence Answer is 5
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Help me with Kudos if it helped you "

Mathematics is a thought process.

gmatclubot

Re: When 51^25 is divided by 13, the remainder obtained is:
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19 May 2014, 09:37

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