If 2n = 7m, then 2n and 7m are the same number, so have the same divisors. So 2n is divisible by 7, and thus n is divisible by 7, and 7m is divisible by 2, which means m is divisible by 2. But it's possible, say, that n=7 and m=2, in which case their GCD is 1, or it could be that n=700 and m=200, and then their GCD is 100, among many other possibilities.

Once we know from Statement 1 that m is prime, then if 2n = 7m, it must be true that m=2, and thus n=7, so their GCD is 1 and the answer is C.

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

Important aside: 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. Similarly, if m=11, then n=77/2. For n to be an integer, m must be even.

If m must be even, it could be the case that m=2 and n=7, in which case the GCD=1
Or it could be the case that m=4 and n=14, in which case the GCD=2
Or it could be the case that m=10 and n=35, in which case the GCD=5
And so on.
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

Cheers,
Brent