GCF problem

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GCF problem

by Nijo » Fri Aug 15, 2014 8:36 pm
What is the GCF of m and n?
1) m is prime
2) 2n = 7m

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by Brent@GMATPrepNow » Fri Aug 15, 2014 8:57 pm
Here's the official wording:
What is the greatest common divisor of positive integers m and n?

1) m is a prime number
2) 2n = 7m
Target question: What is the GCD of m and n?

Statement 1: m is a prime number
If m is a prime number, it has exactly 2 divisors (1 and m), so this tells us that the GCD of m and n must be either 1 or m.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT.

Statement 2: 2n = 7m
If 2n = 7m then we can rearrange the equation to get n = (7/2)m

IMPORTANT: Notice that if m were to equal an ODD number, then n would not be an integer. For example, if m = 3, then n = 21/2 (n is not an integer). Similarly, if m = 11, then n = 77/2 (n is not an integer). So, in order for n to be an INTEGER, m must be EVEN.

If m must be EVEN, there are several possible values for m and n. Consider these two cases:
case a: m = 2 and n = 7, in which case the GCD = 1
case b: m = 4 and n = 14, in which case the GCD=2
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT.

Statements 1 & 2 combined
From statement 1, we know that m is prime, and from statement 2, we know that m is even.
Since 2 is the only even prime number, we can conclude that m must equal 2.
If m = 2, then n must equal 7, which means that the GCD must be 1.
Since we are able to answer the target question with certainty, statements 1 & 2 combined are sufficient, and the answer is C

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by GMATGuruNY » Sat Aug 16, 2014 2:28 am
What is the greatest common divisor of positive integers m and n?

1) m is a prime number
2) 2n = 7m
Statement 1 is clearly INSUFFICIENT.
When one of the statements is clearly insufficient, consider how it might affect the OTHER statement.
Since statement 1 is in terms of m, rephrase statement 2 in terms of m.

Statement 2: m = (2/7)n
The smallest possible value for n is 7.

Case 1: n=7, m = (2/7)(7) = 2.
The GCF of 2 and 7 is 1.

Case 2: n=14, m=(2/7)(14) = 4.
The GCF of 4 and 14 is 2.

Since the GCF can be different values, INSUFFICIENT.

Statements combined:
Cases 1 and 2 imply that m must be EVEN.
One more case to confirm:

Case 3: n=21, m = (2/7)(21) = 6.
Case 3 confirms that m must be even.

Since statement 1 indicates that m is prime, only Case 1 -- m=2, n=7 -- satisfies both statements.
In Case 1, the GCF m and n is 1.
SUFFICIENT.

The correct answer is C.
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