I'm having problems trying to identify the error in this proof in the question below:
Let $u$, $m$, $n$ be three integers. If $u\mid mn$ and $\gcd(u,m) = 1$, then $m = \pm1$.
- If $\gcd(u,m) = 1$, then $1 = us + mt$ for some integers $s$, $t$.
- If $u\mid mn$, then $us = mn$ for some integer $s$.
- Hence, $1 = mn + mt = m(n + t)$, which implies that $m\mid1$, and therefore $m = \pm1$.
My thought is the error is between steps 2 and 3. Inferring that $us = mn$ for some integer $s$ is correct. But substituting into the formula is incorrect because we now have .. two different "$m$" integers in the final step.
I can easily find a counterexample, but I'm not sure if the reasoning for my identifying the error is correct.