Question

Prove that any number n of the form abcabc, where a, b, and c are decimal digits, is divisible by 7, 11 and 13.

Answer 1

We can factorize abcabc as

abcabc = abc000 + abc

abcabc = abc (1000 + 1)

abcabc = abc × 1001

abcabc = abc × 7 × 11 × 13

so naturally each of 7, 11, and 13 must divide abcabc.