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If n and m are positive integers, is m a factor of n?

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by NandishSS » Fri Jan 13, 2017 6:20 am
If n and m are positive integers, is m a factor of n?
(1) n = 5(3^k), for any positive integer k
(2) m = (3)^k-1, for any positive integer k

OA:C

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Source: — Data Sufficiency |
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by GMATGuruNY » Fri Jan 13, 2017 8:00 am
NandishSS wrote:If n and m are positive integers, is m a factor of n?
(1) n = 5(3^k), for any positive integer k
(2) m = (3)^k-1, for any positive integer k
Question stem, rephrased:
Does n/m = integer?

Clearly, each statement on its own is insufficient to know whether n/m = integer.
Statements combined:
n/m = (5 * 3^k) / 3^(k-1) = 5 * 3^(k-(k-1)) = 5*3 = 15.
Thus, n/m = integer.
SUFFICIENT.

The correct answer is C.
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by [email protected] » Fri Jan 13, 2017 10:22 am
Hi NandishSS,

This question is built around a subtle exponent rule "pattern", but if you don't recognize that pattern, then you can still get the correct answer (and prove that the pattern exists) by TESTing VALUES.

We're told that M and N are POSITIVE INTEGERS. We're asked if M is a factor of N. This is a YES/NO question.

1) N = 5(3^K), for any positive integer K

This Fact tells us NOTHING about M, so there's no way to know whether M is a factor of N or not.
Fact 1 is INSUFFICIENT

2) M = (3)^(K-1), for any positive integer K

This Fact tells us NOTHING about N, so there's no way to know whether M is a factor of N or not.
Fact 2 is INSUFFICIENT

Combined, let's TEST VALUES...

IF....
K = 1
N = 5(3) = 15
M = 3^0 = 1
The answer to the question is YES

IF....
K = 2
N = 5(9) = 45
M = 3^1 = 3
The answer to the question is YES

IF....
K = 3
N = 5(27) = 135
M = 3^2 = 9
The answer to the question is YES

Notice that both N and M consistently "triple" from TEST to TEST? This means that M will ALWAYS be a factor of N and the answer ALWAYS YES.
Combined, SUFFICIENT

Final Answer: C

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by Jeff@TargetTestPrep » Thu Jan 19, 2017 5:12 pm
NandishSS wrote:If n and m are positive integers, is m a factor of n?
(1) n = 5(3^k), for any positive integer k
(2) m = (3)^k-1, for any positive integer k
We are given that n and m are positive integers and need to determine whether m is a factor of n, i.e., whether n/m = integer.

Statement One Alone:

n = 5(3^k), for any positive integer k

Since we do not have any information regarding m, statement one alone is not sufficient to answer the question.

Statement Two Alone:

m = 3^(k-1), for any positive integer k

Since we do not have any information regarding n, statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using statements one and two, we can create the following equation:

n/m = 5(3^k)/3^(k-1)

n/m = 5(3^k)/(3^k)(3^-1)

n/m = 5/(3^-1)

n/m = 5 x 3 = 15

Thus, n/m IS an integer.

Answer: C

Jeffrey Miller
Head of GMAT Instruction
[email protected]

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by Matt@VeritasPrep » Fri Jan 20, 2017 12:22 am
It might be easier to see with proper exponents and some color coding:

Given S1 and S2, we've got

n/m = 5 * 3� / 3��¹

m is a factor of n if and only if n/m is an integer. Since 3� / 3��¹ = 3, we can replace that in our equation above, giving

n/m = 5 * 3

and showing that n/m is an integer, making m a factor of n.
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