Divisibility Rules

Quick checks that tell you whether a number divides evenly by 2, 3, 4, 5, 6, 8, 9, or 10, using its digits instead of doing the division.

b a a ÷ b leaves remainder 0

A quick reference: the test to apply for each common divisor. The rules use a number's digits, not division.

What each symbol means

• a: the number you are testing (for example 348)
• b: the divisor you are checking against (for example 3)
• b a: read "b divides a": a divides evenly by b, with nothing left over

A number is "divisible by" another when it splits evenly, leaving no remainder. Divisibility rules are fast checks for this. They let you tell at a glance whether a number can be divided by 2, 3, 4, 5, 6, 8, 9, or 10, without reaching for long division.

The rules, one by one

Each rule looks at a different feature of the number: its last digit, its last two digits, or the sum of all its digits.

• **Divisible by 2:** the last digit is even, that is 0, 2, 4, 6, or 8. Example: 3,706 ends in 6, so it is divisible by 2.
• **Divisible by 3:** the sum of the digits is divisible by 3. Example: 4,821 has digit sum 4+8+2+1 = 15, and 15 ÷ 3 = 5, so yes.
• **Divisible by 4:** the number formed by the last two digits is divisible by 4. Example: 7,516 ends in 16, and 16 ÷ 4 = 4, so yes.
• **Divisible by 5:** the last digit is 0 or 5. Example: 2,340 ends in 0, so it is divisible by 5.
• **Divisible by 6:** it passes the test for 2 AND the test for 3 (because 6 = 2 × 3). Example: 4,821 is odd, so it fails, it is not divisible by 6.
• **Divisible by 8:** the number formed by the last three digits is divisible by 8. Example: 12,120 ends in 120, and 120 ÷ 8 = 15, so yes.
• **Divisible by 9:** the sum of the digits is divisible by 9. Example: 6,381 has digit sum 6+3+8+1 = 18, and 18 ÷ 9 = 2, so yes.
• **Divisible by 10:** the last digit is 0. Example: 4,570 ends in 0, so it is divisible by 10.

If the digit sum is still large, add again: For the 3 and 9 tests, if the digit sum itself is big, add its digits once more. For 8,973: 8+9+7+3 = 27, then 2+7 = 9. Since 9 is divisible by both 3 and 9, so is 8,973.

Rules stack for bigger divisors: You can combine rules. A number is divisible by 12 when it passes both the 4 test and the 3 test, because 12 = 4 × 3 and 4 and 3 share no common factor. The same idea builds tests for 15 (3 and 5), 18 (2 and 9), and more.

A worked example

Is 4,821 divisible by 3? By 9?

1. Both the 3 test and the 9 test use the digit sum, so add the digits once: 4 + 8 + 2 + 1 = 15.
2. For 3: is 15 divisible by 3? Yes, 15 ÷ 3 = 5. So 4,821 is divisible by 3.
3. For 9: is 15 divisible by 9? No, 9 does not divide 15 evenly. So 4,821 is NOT divisible by 9.
4. The digit sum answered both questions at once.

• For 2, 5, and 10, look only at the last digit.
• For 4 and 8, look at the last two or three digits.
• For 3 and 9, add the digits and test that sum.
• For 6 (and other composite divisors), combine the rules of the factors.

Why each kind of rule works

1. LAST-DIGIT rules (2, 5, 10): every place value except the ones is a multiple of 10, and 10 is already divisible by 2, 5, and 10. So only the ones digit can affect divisibility by 2, 5, or 10, which is why we look at it alone.
2. DIGIT-SUM rules (3, 9): each place value is one more than a multiple of 9 (10 = 9+1, 100 = 99+1, 1000 = 999+1). The 9-parts are divisible by both 3 and 9, so the only leftover is the sum of the digits. The whole number is divisible by 3 or 9 exactly when that digit sum is.
3. LAST-TWO / LAST-THREE-DIGIT rules (4, 8): 100 is divisible by 4, and 1000 is divisible by 8. So everything above the last two digits is already a multiple of 4, and everything above the last three digits is already a multiple of 8. Only the trailing digits decide the test.
4. COMBINED rules (6, 12, 15, ...): if a number passes the tests for two factors that share no common factor (like 2 and 3 for 6), it must be divisible by their product. That is why "divisible by 6" is simply "divisible by 2 and by 3".

Practice questions

Question 1

Is 250 divisible by 5?

1. The rule for 5: check the last digit.
2. The last digit of 250 is 0.
3. A last digit of 0 or 5 passes the test.
4. So 250 is divisible by 5 (indeed 250 ÷ 5 = 50).

Answer: Yes

Question 2

Is 471 divisible by 3?

1. The rule for 3: add the digits and test the sum.
2. Add: 4 + 7 + 1 = 12.
3. Is 12 divisible by 3? Yes, 12 ÷ 3 = 4.
4. So 471 is divisible by 3 (indeed 471 ÷ 3 = 157).

Answer: Yes

Question 3

Is 536 divisible by 4?

1. The rule for 4: look only at the last two digits.
2. The last two digits of 536 form the number 36.
3. Is 36 divisible by 4? Yes, 36 ÷ 4 = 9.
4. So 536 is divisible by 4.

Answer: Yes

Question 4

Is 738 divisible by 9?

1. The rule for 9: add the digits and test the sum.
2. Add: 7 + 3 + 8 = 18.
3. Is 18 divisible by 9? Yes, 18 ÷ 9 = 2.
4. So 738 is divisible by 9.

Answer: Yes

Question 5

Is 124 divisible by 6?

1. The rule for 6: the number must pass BOTH the 2 test and the 3 test.
2. Test for 2: the last digit is 4, which is even, so it passes the 2 test.
3. Test for 3: the digit sum is 1 + 2 + 4 = 7, and 7 is not divisible by 3, so it fails the 3 test.
4. Because it fails one of the two tests, 124 is not divisible by 6.

Answer: No