Divisibility rules for 2, 3, 4, 5, 6, 9 and 10

Divisibility rules let you tell whether a whole number divides evenly by 2, 3, 4, 5, 6, 9 or 10 just by looking at its digits. For 2, 5 and 10, check the last digit. For 4, check the last two digits. For 3 and 9, add up the digits. For 6, check the rules for 2 and 3 together.

Let us start with a small puzzle.

A puzzle to begin

A hypothetical baker has 2,736 biscuits and wants to pack them into boxes of 9 with none left over. Can she?

Add the digits: 2 + 7 + 3 + 6 = 18. Eighteen is a multiple of 9, so yes. In fact 2,736 divided by 9 is 304, and you can check it: 304 times 9 is 2,736. No long division was needed to know the answer.

Last-digit rules: 2, 5 and 10

• Divisible by 2: the last digit is 0, 2, 4, 6 or 8. So 58 is divisible by 2 (58 divided by 2 is 29), and 731 is not, because it ends in 1.
• Divisible by 5: the last digit is 0 or 5. So 2,345 is divisible by 5 (2,345 divided by 5 is 469).
• Divisible by 10: the last digit is 0. So 8,970 is divisible by 10 (8,970 divided by 10 is 897).

Why only the last digit? Any whole number is some number of tens plus its last digit. For example, 2,345 is 234 tens plus 5. Ten is divisible by 2, by 5 and by 10, so the tens part never causes trouble. Only the last digit can.

The last-two-digits rule: 4

A number is divisible by 4 when the number formed by its last two digits is divisible by 4.

• 1,316 ends in 16, and 16 is divisible by 4, so 1,316 is too. Check: 1,316 divided by 4 is 329.
• 1,318 ends in 18, and 18 is not divisible by 4, so 1,318 is not. Check: 4 times 329 is 1,316, which leaves a remainder of 2.

The reason is the same idea one step further. Every whole number is some number of hundreds plus its last two digits, and 100 is divisible by 4, because 4 times 25 is 100.

Digit-sum rules: 3 and 9

• Divisible by 3: the sum of the digits is divisible by 3. For 4,152, the sum is 4 + 1 + 5 + 2 = 12, and 12 is divisible by 3, so 4,152 is too. Check: 4,152 divided by 3 is 1,384.
• Divisible by 9: the sum of the digits is divisible by 9. For 81,729, the sum is 8 + 1 + 7 + 2 + 9 = 27, which is divisible by 9. Check: 81,729 divided by 9 is 9,081.

Notice that 4,152 passes the 3 test but not the 9 test, because 12 is not a multiple of 9. If the digit sum is still large, add its digits again: 27 becomes 2 + 7 = 9.

Why does adding digits work? Because 10 is 9 + 1, 100 is 99 + 1, and 1,000 is 999 + 1. Take 352. It equals 3 hundreds plus 5 tens plus 2, which we can rewrite as 3 times 99, plus 5 times 9, plus 3 + 5 + 2. The parts with 99 and 9 are multiples of 9, and so of 3 as well. All that is left to decide the question is 3 + 5 + 2, which is 10. Ten leaves a remainder of 1 when divided by 9, and so does 352: 9 times 39 is 351.

The combined rule: 6

Six is 2 times 3, and 2 and 3 share no common factor other than 1. So a number is divisible by 6 exactly when it passes both the 2 rule and the 3 rule.

• 4,152 is even, and its digit sum, 12, is divisible by 3. So it is divisible by 6. Check: 4,152 divided by 6 is 692.
• 4,155 has digit sum 15, so it is divisible by 3, but it is odd. So it is not divisible by 6.

One warning: this trick only works when the two factors share no common factor. Passing the 2 and 4 tests does not mean a number is divisible by 8. The number 12 passes both, yet 12 divided by 8 is 1.5.

The rules on one page

• 2: last digit even.
• 3: digit sum divisible by 3.
• 4: last two digits divisible by 4.
• 5: last digit 0 or 5.
• 6: passes the rules for 2 and 3.
• 9: digit sum divisible by 9.
• 10: last digit 0.

A variation for you

Twelve is 3 times 4, and 3 and 4 share no common factor, so a number is divisible by 12 when it passes both the 3 and 4 tests. Try 1,764: its last two digits, 64, are divisible by 4, and its digit sum is 18. So 1,764 divided by 12 should be a whole number, and it is: 147.

Now one for you. Is 5,832 divisible by 4, 6, 9 and 12? Work it out before reading on.

The answer: yes to all four. The last two digits, 32, are divisible by 4. The number is even and the digit sum is 18, so it passes 6 and 9. It passes 3 and 4, so it passes 12. The quotients are 1,458, 972, 648 and 486. Multiply each back to check.