Digit sum & divisibility rules (2 through 13) explained

Divisibility rules let you check whether a number divides evenly into another without doing the division at all — just by looking at its digits. In exams, this turns a 30-second long-division check into a 3-second glance. Here's every rule from 2 through 13, with the reasoning behind each so they actually stick.

2 — check the last digit

A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).

3 — sum the digits

A number is divisible by 3 if the sum of its digits is divisible by 3.

This works because 10 leaves a remainder of 1 when divided by 3, so each digit's "place value contribution" to the remainder is just the digit itself — the whole number's remainder mod 3 equals its digit sum's remainder mod 3.

4 — check the last two digits

A number is divisible by 4 if its last two digits form a number divisible by 4.

This works because 100 is divisible by 4, so everything except the last two digits contributes nothing to the remainder.

5 — check the last digit

A number is divisible by 5 if its last digit is 0 or 5.

6 — combine the rules for 2 and 3

A number is divisible by 6 if it passes both the divisible-by-2 and divisible-by-3 checks, since 6 = 2 × 3.

7 — double-and-subtract the last digit

Drop the last digit, double it, and subtract that from the remaining number. If the result is divisible by 7 (including 0), so is the original. Repeat for large numbers.

This rule is the least intuitive one on this list and slower to apply than the others, so it's often faster in an exam to just test-divide by 7 directly for 2–3 digit numbers rather than use this rule.

8 — check the last three digits

A number is divisible by 8 if its last three digits form a number divisible by 8.

Same logic as the rule for 4, extended one more place: 1000 is divisible by 8, so only the last three digits matter.

9 — sum the digits (like 3, but stricter)

A number is divisible by 9 if the sum of its digits is divisible by 9. Same reasoning as the rule for 3, just a tighter divisor.

10 — check the last digit

A number is divisible by 10 if it ends in 0. This one needs no explanation, but it's worth stating for completeness since it's used constantly in combination rules.

11 — alternating digit sum

Add and subtract digits alternately from right to left. If the result (including 0 or negative values treated as their absolute value) is divisible by 11, so is the number.

This works because 10 ≡ −1 (mod 11), so each digit's place value alternates between contributing +1 and −1 to the remainder as you move left.

12 — combine the rules for 3 and 4

A number is divisible by 12 if it passes both the divisible-by-3 and divisible-by-4 checks, since 12 = 3 × 4.

13 — similar to 7, less commonly tested

Multiply the last digit by 4 and add it to the rest of the number. If the result is divisible by 13, so is the original. Repeat as needed.

Like the rule for 7, this one is rarely faster than direct division for exam-sized numbers, so it's more useful to know it exists than to drill it heavily.

Combination rules are the real time-savers

The biggest speed gain isn't any single rule — it's combining them. Checking divisibility by 6, 12, 15, 18, or similar composite numbers by breaking them into coprime factors (2×3, 3×4, 3×5, 2×9) turns one hard check into two easy ones.

Where this actually helps

Divisibility checks show up directly in simplification and number-series questions, and indirectly whenever you need to factor a number quickly — for example, to simplify a fraction or spot a common factor before doing a bigger calculation. A quick digit-sum glance can save you from a multiplication or division you didn't actually need to do.