An error in a proof due to variable creep?

Question

I'm working through some exam practice questions and I came across this one:

Identify the error in the following “proof.”

Let u, m, n be three integers. If u|mn and gcd(u, m) = 1, then m = ±1. If gcd(u, m) = 1, then 1 = us + mt for some integers s, t. If u|mn, then us = mn for some integer s. Hence, 1 = mn + mt = m(n + t), which implies that m|1, and therefore m = ±1.

Now I think the problem here is that we cannot go from the linear transformation statement 1 = us + mt and then state that u|mn => mn = us for some integer s, because we already have a value of s in the previous statement. so we would need to define u|mn => mn = uj for some integer j. And now the substitution doesn't work and we cannot proceed.

Does this make sense to you guys? It makes sense to me that we cannot use s in two different places here. But I'm having a hard time trying to explain why in a clear manner.

Answer 1

You're right: the "proof" has defined $s$ in incompatible ways in two different places. You could explain it by analogy as follows:

We prove that $2=4$. Indeed, the number $2$ is even, so it can be expressed as $2 = 2n$ for some $n$. The number $4$ is even, so it can be expressed as $4 = 2n$ for some $n$. Hence $2 = 2n = 4$, so $2 = 4$.