Step 1 of 4 · Why each rule works
Each rule lets you skip long division. Here's why.
374
=
370
+
4
370 = 37 × 10 — and 10 is divisible by 2, 5, and 10. So 370 is too.
Only the last digit (4) decides. 4 is even → 374 ÷ 2 ✓ · not 0/5 → ÷5 ✗ · not 0 → ÷10 ✗
156
→
1 + 5 + 6 = 12
10, 100, 1000, ... all leave remainder 1 when divided by 3 or 9 (since 10 = 9+1).
So 156 mod 3 = (1 + 5 + 6) mod 3 = 12 mod 3 = 0 ✓
Same trick for 9: digit sum 12, 12 mod 9 = 3 — so 156 is NOT ÷9.
If a number is divisible by both 2 and 3, it's divisible by their product, 6. (Works because 2 and 3 share no common factors.)