If m and n are nonzero integers, is m/n an integer?
(1) 2m is divisible by n
(2) m is divisible by 2n

Let's rephrase this target question.
If m/n is an integer, then we know that m = nk where k is an integer.
So, let's rephrase this target question as: Does m = kn for some integer k?

Statement 1: 2m is divisible by n
This means that 2m = qn for some integer q.
Divide both sides by 2 to get: m = (q/2)n
Do we have enough information to answer the target question? In other words, Does m = kn for some integer k?
Since (q/2) may or may not be an integer, we do not have sufficient information.

Statement 2: m is divisible by 2n
This means that m = q(2n) for some integer q.
Or we can write: m = 2qn
Do we have enough information to answer the target question? In other words, Does m = kn for some integer k?
Yes! Since q is an integer, we know that (2q) must be an integer.
So, we can be certain that m = kn for some integer k?
As such, statement 2 is sufficient, and the answer is B.

Cheers,
Brent

The other option here is to TEST NUMBERS, trying to prove insufficiency.

Target question: If m and n are nonzero integers, is m/n an integer?

(1) 2m is divisible by n

ex.: m = 6 and n = 3 --> this obeys the statement, because (2)(6) is divisible by 3. 6 is divisible by 3, so this gives us a "YES" answer to the question.

Now try to come up with an example that keeps the statement true but gives us a "NO" answer...

ex.: m = 3 and n = 6 --> this obeys the statement, because (2)(3) is divisible by 6. However, 3 is NOT divisible by 6, so this gives us a "NO" answer to the question.

Insufficient.

(2) m is divisible by 2n

ex.: m = 6 and n = 3 --> this obeys the statement, because 6 is divisible by (2)(3). 6 is divisible by 3, so this gives us a "YES" answer to the question.

Now try to come up with an example that keeps the statement true but gives us a "NO" answer...

We can't! If m is divisible by 2n, it must be divisible by both component pieces: 2 and n. There are no numbers we could come up with that would keep the statement true but give us a "NO" answer to the question.

Sufficient. The answer is B.

For a more complex variation on the same idea, try DS #149 in OG 2016:

A school administrator will assign each student in a group of n students to one of m classrooms. If 3 < m < 13 < n, is it possible to assign each of the n students to one of the m classrooms so that each classroom has the same number of students assigned to it?

(1) It is possible to assign each of 3n students to one of m classrooms so that each classroom has the same number of students assigned to it.

(2) It is possible to assign each of 13n students to one of m classrooms so that each classroom has the same number of students assigned to it.