Let $n=5k$, $n$ and $k$ are integers. I will assume $n+3$ is divisible by $5$ which means there is an $m$ such that $n+3=5m$.
Now, $n+3-3=5m-3$, i.e. $n=5m-3$. We know that $n$ is divisible by $5$ so this can be $5k=5m-3$ therefore $3/5=(m-k)$, but this is a contradiction because $m$ and $k$ are integers.