Divisibility of integer numbers If we have two integers $a$ and $b$ such that $a = \frac{5b}{6}$, is $a$ divisible by $5$?

If so, why is that?

I don't see it.
No prime factorisation is really required here: we have $5b=6a$,so $5$ divides $6a$. Now $5$ is prime, hence  by Euclid's lemma, it must divide one of the factors. As it does not divide $6$, it must divide $a$.
If $a$ is an integer then $b = 6k, k\in \mathbb{Z}$ so $$a = 5k \Rightarrow a \equiv 0 \pmod{5}$$