Here's another howler some people commit: If m, n are integers such that m divides n^2 then m divides n.
It's true sometimes, for example if m is prime (or more generally squarefree, i.e. a product of distinct primes). But in general all one can conclude is that there exists integers p, q, r with p squarefree such that $ m = p q^2 $ and $ n = p q r $
The usual counterexample is that 8 divides 4^2 but not 4 ;-)