Find all pairs $a, b \in \mathbb{N}$ such that

$$
a(a+5)=b(b+1)
$$
We immediately notice the obvious solution $a=b=0$. Now we assume $a, b>0$. If $b \leq a$, then $(b+1)<(a+5)$ and $a(a+5)<b(b+1)$. We make the same reasoning with $b+1 \geq a+5$, so $a+1 \leq b \leq a+3$, and there are 3 cases to check:

\[
\begin{aligned}
& b=a+1: \quad a(a+5)=(a+1)(a+2) \Leftrightarrow a^{2}+5 a=a^{2}+3 a+2 \Leftrightarrow a=1 \\
& b=a+2: \quad a(a+5)=(a+2)(a+3) \Leftrightarrow a^{2}+5 a=a^{2}+5 a+6 \Leftrightarrow 0=6 \\
& b=a+3: \quad a(a+5)=(a+3)(a+4) \Leftrightarrow a^{2}+5 a=a^{2}+7 a+12 \Leftrightarrow a=-6
\end{aligned}
\]

The only integer solutions are therefore $(0,0)$ and $(1,2)$.