Confusion while checking own proofs

Question

I am currently studying the book 'How to Think Like a Mathematician' and working on Part 5. I have some confusion regarding the process of checking my own proofs. Sometimes, I feel that a particular step is incorrect, but I cannot provide a solid reason why, nor can I explain why it is correct. For example, in the proof of 'If m²|n², then m|n,' I have proven it as follows: By assuming the existence of k² ∈ ℤ such that n² = m²k², we can conclude that n = mk, which means that m divides n. However, I have a feeling that taking k² ∈ ℤ might be incorrect, but I don't know why, nor do I have a solid reason for why it is correct.

Answer 1

I can't answer the general question about what is a suitable learning practice; that will depend on the person and on the goals.

But I can answer the specific question about your proof:

If $m,n\in\Bbb Z$ have $m^2|n^2$, then say $k\in\Bbb Z$ with $n^2=m^2k^2$ and thus $n=mk$, so $m|n$.

There are two mistakes here. One is important, the other is not. The unimportant mistake is going from: $n^2=m^2k^2$ and deducing: $n=mk$. It may be that $n=-mk$, for all you know. But in that case it would still be true that $m|n$, so it's not a big deal (you should still watch out for it though: it might have been a big deal).

The slightly more serious mistake is saying that there should be $k\in\Bbb Z$ with $n^2=m^2k^2$. You were right to feel uneasy about this step. In general, I would advise seriously asking yourself why is that true before you write it down: it's easy for carelessness to creep in, so self-checking can be helpful.

$m^2|n^2$ tells you only that there is a (nonnegative) integer $\lambda$ with $n^2=m^2\lambda$. But, a priori, it's not necessarily the case that $\sqrt{\lambda}\in\Bbb Z$: suppose, for instance, that $n^2=3m^2$. As $\sqrt{3}$ is not an integer (or even a rational) your proof does not apply.

Hint: is it possible for $n^2=m^2\lambda$ to be true, with all quantities being integers, if $\sqrt{\lambda}\notin\Bbb Z$? Prove or disprove it!