Problem
NT-B1-M10-P020 Difference with the Reversed Number
#20
★★★☆☆ Level 3 of 5
Let \(N\) be a four-digit number, and let \(M\) be obtained by reversing its digits. Prove that \(N-M\) is divisible by \(9\).
The numbers \(N\) and \(M\) have the same digit sum.
The numbers \(N\) and \(M\) have the same digit sum, hence they are congruent modulo \(9\). Therefore \(N-M\equiv0\pmod9\), so \(9\mid N-M\).
One can later discuss divisibility by \(99\) in special cases.