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

I solved it and wrote up my solution.

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.)

They do not. The smallest positive one that does not is $19 \times 11 = 209,$ which has the odd digit sum $2 + 0 + 9 = 11.$