6. Prove that if any three-digit number is written twice in a row to get a six-digit number, it must be divisible by 7, 1 1 and 13 simultaneously.
Let any three numbers be ABC, then six numbers.
ABCABC= 1000×ABC+ABC
= 100 1×ABC
=7× 1 1× 13×ABC
So this six-digit number must be divisible by 7, 1 1 and 13 at the same time.