Divisibility Condition for abbb

The 4-digit number $N = 1000a + 111b$. For $N$ to be divisible by $a$, $1000a + 111b \equiv 0 \pmod a \iff 111b \equiv 0 \pmod a$. Since $111 = 3 \times 37$, $111b = 37 \times 3b$. Since $a$ is a single digit ($1 \le a \le 9$), $a$ is coprime to 37. Therefore, $37 \times 3b$ is divisible by $a$ if and only if $3b$ is divisible by $a$.