Explanation

17 ≡ −1 (mod 18), so 17^29 ≡ (−1)^29 = −1 (mod 18). 19 ≡ 1 (mod 18), so 19^29 ≡ 1. Adding, 17^29 + 19^29 ≡ −1 + 1 = 0 (mod 18), so the remainder on division by 18 is 0.