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.

jain2016 wrote:What is the greatest common divisor of positive integers m and n ?

1) m is a prime number

2) 2n = 7m

Statement 1: m is a prime number

Since m could be any prime number and n could be equal to m, there are many possible answers that satisfy Statement 1.

Insufficient.

Statement 2: 2n = 7m

Since n and m are integers, this statement implies that 7 is a factor of n and 2 is a factor of m, and that other than 7 and 2, the factors of n and m are the same.

In other words, n/m = 7/2, n:m = 7:2, and n = 7k and m = 2k.

For instance, n could be 7 x 5 x 11 and m could be 2 x 5 x 11, in which case their greatest common divisor would be 55.

Alternatively n could be 7 x 1 and m could be 2 x 1, in which case their greatest common divisor would be 1.

Insufficient.

Statements Combined:

There is only one number that is both prime and a multiple of 2. So m = 2 and n = 7, and the greatest common divisor of n and m is 1.

Sufficient.

The correct answer is C.

jain2016 wrote:What is the greatest common divisor of positive integers m and n ?

1) m is a prime number

2) 2n = 7m

We need to determine the greatest common divisor, or the greatest common factor (GCF), of integers m and n.

Statement One Alone:

m is a prime number.

Since we don't know anything about n, statement one is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

2n = 7m

We can manipulate the equation 2n = 7m:

n = 7m/2

n/m = 7/2

Even with the equation rewritten, we see that there are many options for m and n, and thus there are many different GCFs for m and n. For instance, if n = 7 and m = 2, then the GCF is 1. However, if n = 14 and m = 4, then the GCF is 2. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Using statements one and two, we know that m is prime and that n/m = 7/2. Therefore, m must equal 2 and n must equal 7. When m is 2 and n is 7, the GCF is 1.

Answer: C