[Math] Take any three-digit positive integer, reverse its digits, and subtract. The difference is divisible by 11.

Question

Take any three-digit positive integer, reverse its digits, and subtract. For example, 742 − 247 =
495. The difference is divisible by 11. Prove that this must happen for all three-digit numbers abc

Answer 1

\begin{align} x &= 100a + 10b + c \quad &\text{Any three digit number can be expressed in this form}\\ y &= 100c + 10b +a \quad &\text{This is the number once the digits are reversed}\\ x-y &= 100a - a + 10b -10b +c -100c = 99a-99c\\ x-y &=11 \cdot(9a-9c) \end{align}