If x, y, p and q are positive integers, is x^p a factor of y^q?

For x^p to be a factor of y^q, (y^q)/(x^p) must be an integer. This means if we were to break y and x into their prime factors, y must be a multiple of x and q must be greater than or equal to p.

This should make logical sense if you understand your basic divisibility rules and how to cancel out numbers when dividing. Some examples to illustrate:

Example 1: (3^5)/(3^2) = 3^3 = 27(an integer).
Example 2: (9^2)/(3^5) = (3^4)/(3^5) = 1/3(not an integer).

In the second case, note that how even though x is a multiple of y, q is less than p and therefore we don't get an integer.

Statement 1) Using our examples from above we see that in each case x is a factor of y. However in example 1 we get an integer and in example we do not get an integer.

INSUFFICIENT

Statement 2) Since p & q must be integers, this tells us q is greater than or equal to p. This is part of what we're looking for, however we still need to know if y is a multiple of x.

INSUFFICIENT

Statements 1 & 2) Now we know that y is a multiple of x and q is greater than or equal to p. Exactly what we're looking for.

SUFFICIENT

Answer: C