MathNet 2019-part-06 — 184 problems (EN–VI) Question 161

For each four-digit number $a b c d$, we denote by $S$ the number $S = \overline{a b c d} - \overline{d c b a}$. Prove that $S$ is a multiple of $37$ if and only if the two middle digits of the number $a b c d$ are equal.