Prove: If a is prime and a divides k^2, then a divides k?

Question

please provide a written proof explaining how the statement is true.. not a numerical proof or induction. thanks!!

Answer 1

The above does not quite work because it does not rely on the divisor being prime e.g. 8 does not divide 4 but it does divide 16.

I would use a prime decomposition.

if a=p1^n1*p2^n2...
a^2=p1^2n1*p2^2n2...

From this, for pm is a factor of a^2, it means that 2nm is not zero. This means that nm is not zero so pm must also be a factor of a.

Answer 2

Assume that a does not divide k. This means that a cannot divide any factors of k. Then a cannot divide k² since the only factors of k² are also (repeated) factors of k. Therefore if a divides k² it must also divide k.

Doug