Problem Solving

[GMAT math practice question]

If n is an integer, which of the following must also be an integer?

I. n(n+1)/2
II. n(n+1)(n+2)/6
III. n(n+1)(n+2)(n+3)/8

A. I only
B. II only
C. III only
D. I and II
E. I, II and III

=>

Statement I
Since n and n + 1 are consecutive integers, n(n+1) is a multiple of 2.
Thus, n(n+1)/2 is an integer.

Statement II
Since n and n + 1 are consecutive integers, n(n+1) and n(n+1)(n+2) are multiples of 2.
Since n, n + 1 and n + 2 are three consecutive integers, n(n+1)(n+2) is a multiple of 3.
Thus, n(n+1)(n+2) is a multiple of 6, and n(n+1)(n+2) / 6 is an integer.

Statement III
Since n and n + 1 are two consecutive integers, n(n+1) is a multiple of 2.
Similarly, (n+2)(n+3) is a multiple of 2.
Also, either n and n + 2 or n + 1 and n + 3 are consecutive even integers. Thus, either (n + 1)(n+3) is a multiple of 8 or n(n+2) is a multiple of 8 since one of them is a multiple of 4.
It follows that n(n+1)(n+2)(n+3) is a multiple of 8, and n(n+1)(n+2)(n+3)/8 is an integer.

Therefore, the answer is E.

Answer: E