On Integers Whose Sum is the Reverse of their Product

Abstract

We determine all pairs of positive integers $(a,b)$ such that $a+b$ and $a \times b$ have the same decimal digits in reverse order: \[ (2,2), (9,9), (3,24), (2,47), (2,497), (2,4997), (2,49997), \ldots \] We use deterministic finite automata to describe our approach, which naturally extends to all other numerical bases. Our automata are a variation on the notion of Young graphs, which were introduced by Sloane to study ``reverse multiples''.