Question

Show that any number of the form ABCABC (e.g. 356356) is divisible by 13.

Answer 1

We could write such a number as

$10^5a+10^4b+10^3c+10^2a+10b+c$

$=10^2(10^3+1)a+10(10^3+1)b+(10^3+1)c$

$=(10^3+1)(100a+10b+c)$

$10^3+1=1001=13\cdot77$ so we're done.