Divisibillity discret mathmatic

Question

4. Is it all integers n true that if 10 |n^(2) it holds 10|n?

Is it all integers n true that if 9 | n^(2) it holds 9 | n?

How do you prove this does i take different integers to get whole numbers?

Answer 1

If the highest power of prime $p$ divides $n$ is $d$

the highest power of prime $p$ divides $n^2$ is $2d$

if the highest power of $p$ that divides $n$ is $1,$ we $2d\ge1, d\ge1$

Now $10=2^15^1$ where $2,5$ are primes, $10|n\iff 10|n^2$

If $9=3^2$ divides $n^2,2d\ge2\implies d\ge1$

For example, $n=3$