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
Source:Math Revolution
If n and m are positive integers, is m a factor of n?
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Question stem, rephrased: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
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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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
GMAT assassins aren't born, they're made,
Rich
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
GMAT assassins aren't born, they're made,
Rich
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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.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
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
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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.
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.