The Complete Guide to Divisibility Rules (2 Through 13)

Can you tell whether 4,176 is divisible by 8 without a calculator? What about whether 918,082 divides evenly by 11? These questions come up constantly in math class — in simplifying fractions, finding common denominators, factoring, and prime factorization.

The good news: there's a quick test for every number from 2 through 13. Once you learn them, you'll be able to spot divisibility instantly. Here's the complete set.

Divisible by 2

Rule: The last digit is even (0, 2, 4, 6, or 8).

• 4,826 → last digit is 6 (even) → yes
• 7,351 → last digit is 1 (odd) → no

This is the simplest rule. You already know it. It's also the foundation for the rules for 4, 6, 8, and 12.

Divisible by 3

Rule: Add up all the digits. If that sum is divisible by 3, so is the original number.

• 729 → 7 + 2 + 9 = 18 → 18 ÷ 3 = 6 → yes
• 845 → 8 + 4 + 5 = 17 → 17 ÷ 3 = 5 R2 → no
• 123,456 → 1 + 2 + 3 + 4 + 5 + 6 = 21 → 21 ÷ 3 = 7 → yes

If the digit sum is still large, keep summing. For 123,456,789: 1+2+3+4+5+6+7+8+9 = 45 → 4+5 = 9 → divisible by 3.

Divisible by 4

Rule: Look at just the last two digits. If that two-digit number is divisible by 4, so is the whole number.

• 4,176 → last two digits are 76 → 76 ÷ 4 = 19 → yes
• 3,518 → last two digits are 18 → 18 ÷ 4 = 4 R2 → no

Why it works: 100 is divisible by 4, so any hundreds, thousands, etc. are automatically divisible by 4. Only the last two digits matter.

Divisible by 5

Rule: The last digit is 0 or 5.

• 7,835 → ends in 5 → yes
• 4,862 → ends in 2 → no

Divisible by 6

Rule: The number must pass both the test for 2 AND the test for 3.

• 714 → even? yes (4). Digit sum? 7+1+4 = 12, divisible by 3? yes. → yes
• 532 → even? yes (2). Digit sum? 5+3+2 = 10, divisible by 3? no. → no

A number is divisible by 6 only if it's divisible by both of its prime factors (2 and 3). This pattern repeats: any composite number's divisibility can be checked by testing its prime factor components.

Divisible by 7

Rule: Take the last digit, double it, and subtract it from the remaining number. If the result is divisible by 7 (or is 0), so is the original. Repeat if needed.

• 112 → take 2, double it (4), subtract from 11 → 11 − 4 = 7 → yes
• 672 → take 2, double it (4), subtract from 67 → 67 − 4 = 63 → 63 ÷ 7 = 9 → yes
• 905 → take 5, double it (10), subtract from 90 → 90 − 10 = 80 → take 0, double it (0), subtract from 8 → 8 → no

This is the trickiest of the basic divisibility rules, and it's the one most students have never seen. The doubling-and-subtracting process works because of modular arithmetic: 10 ≡ 3 (mod 7), so doubling the last digit and subtracting is equivalent to testing the whole number mod 7.

Divisible by 8

Rule: Look at the last three digits. If that number is divisible by 8, so is the whole number.

• 4,176 → last three digits are 176 → 176 ÷ 8 = 22 → yes
• 23,590 → last three digits are 590 → 590 ÷ 8 = 73 R6 → no

Why it works: 1,000 is divisible by 8, so everything beyond the last three digits is automatically divisible by 8.

Shortcut within the shortcut: If the hundreds digit is even, check if the last two digits are divisible by 8. If the hundreds digit is odd, add 4 to the last two digits and check if that is divisible by 8.

Divisible by 9

Rule: Add up all the digits. If the sum is divisible by 9, so is the original number.

• 729 → 7 + 2 + 9 = 18 → 18 ÷ 9 = 2 → yes
• 843 → 8 + 4 + 3 = 15 → 15 ÷ 9 = 1 R6 → no

Same as the rule for 3, but stricter. Every number divisible by 9 is also divisible by 3, but not vice versa.

Divisible by 10

Rule: The number ends in 0.

• 4,830 → ends in 0 → yes
• 4,835 → ends in 5 → no

Divisible by 11

Rule: Starting from the left, alternately add and subtract each digit. If the result is 0 or divisible by 11, so is the original number.

• 918,082 → 9 − 1 + 8 − 0 + 8 − 2 = 22 → 22 ÷ 11 = 2 → yes
• 1,331 → 1 − 3 + 3 − 1 = 0 → yes
• 5,243 → 5 − 2 + 4 − 3 = 4 → no

This rule surprises most people. The alternating pattern works because powers of 10 alternate between leaving a remainder of 1 and −1 when divided by 11 (10 ≡ −1 mod 11, 100 ≡ 1 mod 11, 1000 ≡ −1 mod 11, etc.).

Divisible by 12

Rule: The number must pass both the test for 3 AND the test for 4.

• 1,044 → digit sum = 1+0+4+4 = 9, divisible by 3? yes. Last two digits = 44, divisible by 4? yes (44÷4=11). → yes
• 648 → digit sum = 18, divisible by 3? yes. Last two digits = 48, divisible by 4? yes (48÷4=12). → yes

Divisible by 13

Rule: Multiply the last digit by 4 and add it to the remaining number. If the result is divisible by 13 (or is 0), so is the original. Repeat if needed.

• 338 → 33 + (8 × 4) = 33 + 32 = 65 → 65 ÷ 13 = 5 → yes
• 1,001 → 100 + (1 × 4) = 104 → 10 + (4 × 4) = 26 → 26 ÷ 13 = 2 → yes

Notice this is the opposite of the rule for 7 (where you subtractdouble the last digit). For 13, you add four times the last digit.

Quick Reference

Practice Challenge

Test yourself: is 5,913,492 divisible by each of the following?

• 2: Last digit is 2 (even) → yes
• 3: 5+9+1+3+4+9+2 = 33, and 33 ÷ 3 = 11 → yes
• 4: Last two digits are 92, and 92 ÷ 4 = 23 → yes
• 6: Divisible by both 2 and 3 → yes
• 9: Digit sum is 33, and 33 ÷ 9 = 3 R6 → no
• 12: Divisible by 3 and 4 → yes

Once you internalize these rules, you'll find yourself checking divisibility automatically — when simplifying fractions, factoring expressions, or just splitting a restaurant bill.

This is Part 2 of our “Math Tricks Your Textbook Never Taught You” series. Next up: Mental Math Shortcuts: How to Calculate Faster Than a Calculator.