GMAT QUANT

Question

When 51^25 is divided by 13, the remainder obtained is: from which book this question was taken?

can anyone please tell me the name of the book

avigutman · Answer

I don’t know, but it doesn’t look like any official GMAT question I’ve ever seen.

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ReedArnoldMPREP · Answer

This would be a pretty difficult question as it tests fairly high level modular arithmetic. Probably higher than the GMAT goes.

It helps to realize 51 is one less than a multiple of 13.

Let's call any multiple of 13 'M13.' So any number 1 less than that is 'M13 - 1'

We have here a situation of raising 'M13 - 1' to a power of 25.

(M13 - 1)*(M13 - 1)*(M13 - 1)...*(M13 - 1)
{25 times}

Consider just (M13 - 1)^2 = (M13 ^ 2 - 2M13 + 1)

Since (M13)^2 and 2(M13) are both multiples of 13, we know (M13 ^ 2 - 2M13 + 1) is 1 more than a multiple of 13. So this can be written as (M13 + 1)

If we multiple (M13+1)*(M13 - 1), well, that's a difference of squares: M13^2 - 1 (...this is a little deceptive, since 'M13' doesn't have to be the *same* multiple of 13, BUT this is beauty of modular arithmetic... you only care about the numbers in relation to the number you're dividing by).

(M13)^2 is another M13, so this is just another (M13 - 1)

*basically*, when we multiply a bunch of 'M13 - 1' together, we'll end up with a multiple of 13 + or - '1.'

Whether it's '+' or '-' depends on the exponent. As you might expect, even exponents will be '+1,' odd will be '-1.'

Since 25 is odd, we'll end up at 'M13 - 1,' which will have a remainder of 12 when divided by 13.

...Again, I don't think I've had to do anything this advanced in the field of modular arithmetic on a GMAT problem in 7 or 8 years of teaching it.

Chinmay1502 · Answer

51^25 = (52-1)^25 = 52^25 - A*52^24 + B*52^23 ...... Z*52^1 - 1

As you can see from the expansion, all the terms except the last term is divisible by 13 as 52 is a multiple of 13.
Thus 1 (last term) is subtracted from a number divisible by 13, which will give us 12 as remainder.

ReedArnoldMPREP , these kinds of questions are pretty common in CAT.

ReedArnoldMPREP · Answer

There are elements similar to this logic that show up in a CAT, but I cannot think of a single *official* GMAT problem that would *require* understanding an expansion like this. If you can point me to one, I'll be glad to amend my statement that this is more advanced than the GMAT requires.

KenAdams1005 · Answer

Sorry, I should have elaborated it. By CAT, I don't mean Computer Adaptive Tests. CAT (Common Admission Test) is an exam through which admissions in MBA colleges happen in India. You can consider it Indian equivalent of GMAT. Only in the sense that both are required for MBA admissions, but the exam pattern and difficulty level is completely different. In this exam these kinds of questions are pretty common. This one can be considered of Easy-Medium difficulty level from perspective of CAT (Common Admission Test).

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Chinmay1502 · Answer

ReedArnoldMPREP

Sorry, I should have elaborated it. By CAT, I don't mean Computer Adaptive Tests. CAT (Common Admission Test) is an exam through which admissions in MBA colleges happen in India. You can consider it Indian equivalent of GMAT. Only in the sense that both are required for MBA admissions, but the exam pattern and difficulty level is completely different. In this exam these kinds of questions are pretty common. This one can be considered of Easy-Medium difficulty level from perspective of CAT (Common Admission Test).

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Posted from my mobile device

ReedArnoldMPREP · Answer

Ahhhh, I see. Thanks for clarifying!

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