Divisibility of the digit-reversal difference (the mirror-number theorem)

We prove that, in any positional base b ≥ 2, the absolute difference between a natural number and its reversal — the number obtained by reversing the order of its digits — is always divisible by b − 1; in particular, in base ten, it is always divisible by 9. The main proof is the author's original one, formulated on 18 May 2003 and based on the digit-sum rule (casting out nines): since a number and its reversal have the same digits, they have the same digit sum and hence the same remainder modulo 9, so their difference is a multiple of 9. A structural form is also given, exhibiting the contribution of each pair of symmetric digits, together with the generalization to arbitrary bases. The result is a classical fact of elementary number theory, closely tied to the divisibility-by-nine criterion; this note is a self-contained exposition with an original proof.