Prove $\forall n \in \mathbb N, \forall k\in \mathbb Z, \forall \ell \in \mathbb Z, \neg (n = 5k+3 \land n = 5\ell +1)$, Intended meaning?

Question

I am understanding this question to prove $5k+3 \neq 5l+1$ for all values of l and k as long as the result is a natural number. Since it's for all, it can easily be disproved by finding any example where it is not equal, e.g. choose k = 5 and l = 100. So I can disprove it by contradiction, did I understand that correctly?

Answer 1

No, that is not correct. You are supposed to prove that no natural number of the form $5k+3$ is also of the form $5l+1$ ($k,l\in\mathbb Z$). So, simply providing an example isn't enough.

That's easy to prove, though: $5k+3=5l+1\implies 5(l-k)=3-1=2$, which is impossible.