Identify the error - Discrete math

Question

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 $m = \pm1$.

1. If $\gcd(u,m) = 1$, then $1 = us + mt$ for some integers $s$, $t$.
2. If $u\mid mn$, then $us = mn$ for some integer $s$.
3. 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.

Answer 1

Firstof all your statement 1 is wrong. Example- 5|2*25, gcd(5,2)=1, but 2 is not equal to 1 or -1. Correct statement should be if u|mn, gcd(u,m)=1 then u|n. And now the error in your proof is you have choosen same integer 's' in both step 2 and 3. 1=us+mt then u|mn implies mn=ua for some integer a.

Answer 2

Look at step 1 when $u=1$, $m=n=2$.