There are several ways to solve this question:

Algebraic approach:

Let n be the smaller integer => 1/n + 1/(n+4) = 10/21
Solve for n => n=3. Hence, answer b) is correct.

Picking numbers:

Start with middle choice, i.e. 5
5 obviously cannot be the larger of the two integers as this would mean that the smaller integer would equal 1 (hence reciprocal also 1) and the sum of the reciprocals would exceed 10/21.
If 5 is the smaller of the two integers => 1/5 + 1/9 = 14/45. This is smaller than 10/21.

Next, choose 3 and 7 (because 12 and 28 would obviously make sum of the reciprocals too small) => 1/3 + 1/7 = 10/21. Hence, answer b) is correct.

Considering Least common multiple:

For the sum of the reciprocals of the integers to have the denominator 21 both numbers must be divisors of 21. This only holds true for answer choice b).

1/n + 1/(n+4) = 1/21 =====> n = 3