Question

For an integer q, an integer x divides q if and only if x divides q^2

is there an integer for which the above statement is true. can you make the statement about which integer x this statement is true

Answer 1

If q/x=p (integer), then q²/x=(q/x)q=pq. So if x is a factor of q then it must be a factor of q².

However, if x is a factor of q², it may not be a factor of q. Example: if x=4 and q=6, q²=36, which is divisible by 4, but 6 is not divisible (exactly) by 4.

If x=a²b, where a is an integer, and q²=a²b² (divisible by x) then q=ab which is not divisible by x (q/x=ab/(a²b)=1/a, which is not an integer).