mod 3: digit sum · mod 11: alternating sum · mod 7 has no trick · 10^k mod m tells you everything · runs locally
trick exists when powers of 10 have a simple pattern mod m · mod 3,9: 10≡1 so digit sum works · mod 11: 10≡−1 so alternating sum works · mod 7: 10³≡6 ≡ −1 so blocks of 3 alternate, but no short rule