费费6-9

9、If n=p/q，and both of p and q are non-zero integers, is n an integer?
(1) n^2 is an integer;

(2) n^3 is an integer;

请问为何答案D?

n^2=(p/q)^2=k,so p=q(k)^1/2. If n^2 is an integer, then, k is an integer. Because integer=integer*integer, if p and q are non-zero integers, (k)^1/2 must be an integer. So p is devisable by q, so n is an integer.

With the same method, if n^3 is an integer, n is an integer.  

“Because integer=integer*integer, if p and q are non-zero integers, (k)^1/2 must be an integer.”---可是分数乘分数也可为整数呀，如2/3*3/2=1，可否再说明白些？Thank you.

n^2 = p^2 / q^3 = p/ q * p / q, if p/q is not an integer then p/q * p/q is not an integer as well, but since p/q * p/q is an integer then that means p/q must be an integer;

the same goes for p^3/q^3