Consider the following groups $3\mathbb Z$ and $5\mathbb Z$ under the addition opertation. Let $h : 3\mathbb{Z}\to 5\mathbb Z$ be such that $h(15)=45$. Can $h$ be a homomorphism?
My approach- For h to be homomorphism, $\forall$ a,b ∈ $3\mathbb Z$ and $\forall$ h(5a/3),h(5b/3) ∈ $5\mathbb Z$ , $h((5a/3)+(5b/3))=h(5a/3)+h(5b/3) $. There I took $a=6$ and $b=9$ and the resulting $h(5a/3)=10$ and $h(5b/3)=15$ which gives $h((5a/3)+(5b/3))=25$.
Is this enough to prove by this counter example that $h$ is not homomorphism?
Hint: $15 = 5\cdot 3$, if $h(15) = 45$, what's $h(3)$? Can it be an element of $5\mathbb Z$?