12 Days of Christmas GMAT Competition - Day 10: If n is a positive two

GMAT Club Official Explanation:

If n is a positive two-digit even integer, which of the following must be a factor of \((n - 5)(n - 3)(n - 1)(n + 1)(n + 3)(n + 5)\)?

I. 11
II. 21
III. 45

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

Since n is a positive two-digit even integer, then \((n - 5)(n - 3)(n - 1)(n + 1)(n + 3)(n + 5)\) is a product of six consecutive odd integers. For consecutive odd integers:

One of every three is a multiple of 3;
One of every five is a multiple of 5;
One of every seven is a multiple of 7;
One of every nine is a multiple of 9;
One of every eleven is a multiple of 11;
And so on.

Hence, out of six consecutive odd integers, we for sure will have two multiples of 3 and at least one multiple of 5. Thus, \((n - 5)(n - 3)(n - 1)(n + 1)(n + 3)(n + 5)\) will always be divisible by 3*3*5 = 45.

However, we cannot guarantee divisibility by either 11 or 7 (thus by 21).

For example, if the six consecutive odd integers are 13, 15, 17, 19, 21, and 23, their product is not divisible by 11, because there is no multiple of 11 among them. Similarly, if the six consecutive odd integers are 9, 11, 13, 15, 17, and 19, their product is not divisible by 7, because there is no multiple of 7 among them.

Answer: C.