## Digit Sums of Multiples of 11

Source

I got this problem from Rustan Leino, who got it from Madan Musuvathi.

I solved it and wrote up my solution.

Problem

Some multiples of 11 have an even digit sum. For example, $7 \times 11 = 77$ and $7+7 = 14,$ which is even; $11 \times 11 = 121$ and $1+2+1 = 4,$ which is even. Do all multiples of 11 have an even digit sum? (Prove that they do or find the smallest positive one that does not.)

Solution