Five-digit numbers starting 37 with rotations divisible by 37

Consider five-digit numbers $\overline{37abc}$ (digits $a,b,c$, with $0$ allowed). Count how many of them have the property that $\overline{37abc}$, $\overline{37bca}$ and $\overline{37cab}$ are each multiples of $37$.

$28$. Since $37\mid37000$, we need $37$ to divide $\overline{abc}$, $\overline{bca}$ and $\overline{cab}$. Because $1000\equiv1\pmod{37}$, $\overline{bca}\equiv10\cdot\overline{abc}$, so one rotation being divisible forces the others. The multiples of $37$ from $000$ to $999$ are $37k$ for $k=0,\dots,27$: $28$ values.