Question

Identify the error:
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 m|1, and therefore m = +-1.

I'm thinking the main error here is that the m = +-1 if gcd(u,m) = 1. u,m, and n could be any number. If gcd(u,m) = 1, then u and m are relative primes, it does not mean that m has to me eitehr 1 or -1. If u = 2 m = 5 n = 6, everything would be true still except for the conclusion m = -+1 would be false. Is this correct?

Answer 1

And is there anything else I am missing about the supposed error here?

Answer 2

" then m = +."
what does this mean?

Answer 3

I'm guessing from what you've written, the first line is the statement: Let u,m,n be three integers. If u|mn and gcd(u,m) = 1 then m = \(\pm\)1. The next three sentences gives a supposed "proof", which is where you are supposed to look for the error. (Obviously the statement is false, as you've spotted.)

The error is to conclude, "Hence, 1 = mn + mt = m(n + t)" based on the previous two sentences. Note that the proof says, "If u|mn, then us = mn for some integer s." This is all true, but the problem is that this integer s may be different from the s, t mentioned in the previous sentence, so you cannot just substitute us.

So that line should say "If u|mn, then ut = mn for some integer t." And now the next line does not follow. That's where the "proof" falls apart.