Digit sums of multiples of 11

Some multiples of 11 have an even digit sum. For example, 7 * 11 = 77 and 7 + 7 = 14, which is even; 11 * 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 that does not.)