Thread: prove that if n is divisible by 11 and n' is obtained from n by inserting two identic

1. prove that if n is divisible by 11 and n' is obtained from n by inserting two identic

prove that if n is divisible by 11 and n' is obtained from n by inserting two identical digits between consecutive digits of n, then n' is also divisible by 11. For example, since 407 is divisible by 11, the following are also divisible by 11: 22407, 43307, 40997 and 40722

Let $n=\sum_{i=0}^md_i10^i$. Use the fact that $10\equiv-1\,(\text{mod}\,\,11)$ and compute mod 11.