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B2.2 · Apply divisibility rules for 2, 3, 4, 5, 6, 8, 9, and 10

Learn to apply divisibility rules for 2, 3, 4, 5, 6, 8, 9, and 10 through clear examples and targeted practice.

Ontario Grade 6 Mathematics

Number

A Grade 6 lesson on checking whether a number divides evenly

Suppose you have 24 counters and want to put the same number in each group, with none left over. Since 24 can be split into 2 equal groups, it divides evenly by 2. A divisibility rule is a quick way to tell whether a whole number divides evenly by another whole number. You do not need to make all the groups to use the rule. This lesson focuses on eight divisors: 2, 3, 4, 5, 6, 8, 9, and 10.

What you will learn

Explain what it means for one whole number to divide another evenly.
Apply the divisibility rules for 2, 3, 4, 5, 6, 8, 9, and 10.
Choose and use a useful rule to check a whole number.

What does “divides evenly” mean?

A whole number is a number such as 0, 1, 2, or 3. A divisor is a number you divide by. When a number divides evenly, the answer is a whole number and there is no remainder. A remainder is an amount left over.

For example, 24 counters can be shared equally among 3 groups. Each group gets 8 counters, and none are left. So 24 divides evenly by 3. In symbols,

24 \div 3 = 8

A divisibility rule lets you check without doing a full division. Some rules look at the last digit. Other rules use the last two or three digits, or add the digits together.

Divides evenly means there is no remainder.
A rule is a quick test for a particular divisor.

Learn the eight rules

Begin with the last digit. A whole number divides evenly by 2 if its last digit is 0, 2, 4, 6, or 8. These are the even digits. It divides evenly by 5 if its last digit is 0 or 5. It divides evenly by 10 if its last digit is 0.

For 4, look at the last two digits as one number. If that number divides evenly by 4, the full number does too. For 8, look at the last three digits as one number. If that number divides evenly by 8, the full number does too. If the full number has fewer than two or three digits, use the whole number.

For 3 and 9, add all the digits. If the digit sum divides evenly by 3, then the original number divides evenly by 3. If the digit sum divides evenly by 9, then the original number divides evenly by 9. A digit sum is the result of adding the digits.

The rule for 6 combines two rules: a number divides evenly by 6 when it divides evenly by both 2 and 3. Check both conditions. Passing only one is not enough.

Last digit: use the rules for 2, 5, and 10.
Last two digits: use the rule for 4.
Last three digits: use the rule for 8.
Digit sum: use the rules for 3 and 9.
For 6, check both 2 and 3.

Choose a rule and check carefully

A number may pass more than one rule. For example, 120 ends in 0, so it divides evenly by 2, 5, and 10. It also divides evenly by 3 because its digit sum is

1 + 2 + 0 = 3

. That means it divides evenly by 6 as well: it passes both the 2 and 3 checks.

When a number fails a rule, identify exactly which test failed. For instance, a number that passes the 2 check but fails the 3 check does not pass the rule for 6. Saying that it is even is not enough to show it divides evenly by 6.

The rules are tests for particular divisors. Do not use the last-digit test for 2 to decide about 4 or 8. Use the part of the number named in that rule.

One number can divide evenly by several of the listed divisors.
To test for 6, both checks must pass.
Match each divisor to its own rule.

A simple routine

First, name the divisor you are testing. Next, use only the matching rule. For a digit-sum rule, add the digits and check that sum. For the rule for 6, apply both the 2 and 3 rules. Finally, state whether the number divides evenly by the chosen divisor.

You can practise by testing one number against several divisors. Keep a record of which tests pass. This helps you avoid treating one successful test as proof for a different divisor.

Name the divisor.
Apply its rule, or both rules for 6.
State the result clearly.

The eight divisibility rules

Divisor	What to check
2	Last digit is 0, 2, 4, 6, or 8.
3	Add all digits; the sum divides evenly by 3.
4	The last two digits divide evenly by 4.
5	Last digit is 0 or 5.
6	The number passes both the 2 and 3 rules.
8	The last three digits divide evenly by 8.
9	Add all digits; the sum divides evenly by 9.
10	Last digit is 0.

Worked example

Testing a number with the last digit and digit sum

Does 3,426 divide evenly by 2, 3, 5, 6, and 9?

Check the last digit
The last digit is 6. It is even, so 3,426 divides evenly by 2. It is not 0 or 5, so the number does not divide evenly by 5.
Add the digits
Add all four digits. The sum is 15. Since 15 divides evenly by 3 but not by 9, the original number divides evenly by 3 but not by 9.
$3 + 4 + 2 + 6 = 15$
Check for 6
The number passes the checks for both 2 and 3. Therefore it divides evenly by 6.

Answer: 3,426 divides evenly by 2, 3, and 6. It does not divide evenly by 5 or 9.

Check: The rule for 6 requires both the 2 check and the 3 check. Both passed.

Worked example

Using the last two and three digits

Does 5,248 divide evenly by 4 and by 8?

Test for 4
Use the last two digits, 48. Since 48 divides evenly by 4, the full number divides evenly by 4.
$48 \div 4 = 12$
Test for 8
Use the last three digits, 248. Since 248 divides evenly by 8, the full number divides evenly by 8.
$248 \div 8 = 31$

Answer: 5,248 divides evenly by both 4 and 8.

Check: The rule for 4 uses two ending digits; the rule for 8 uses three.

Worked example

Checking several rules for a larger number

For 7,035, decide whether it divides evenly by 2, 3, 5, 6, 8, 9, and 10.

Use the last digit
The last digit is 5. The number does not pass the rule for 2 or 10, but it passes the rule for 5.
Use the digit sum
The digit sum is 15. It divides evenly by 3, but not by 9. So 7,035 passes the rule for 3 and fails the rule for 9.
$7 + 0 + 3 + 5 = 15$
Check for 6 and 8
The number fails the 2 check, so it cannot pass the rule for 6, even though it passes the 3 check. For 8, check the last three digits, 035, which is 35. Since 35 does not divide evenly by 8, the number fails the rule for 8. $35 \div 8 = 4$ remainder $3$

Answer: 7,035 divides evenly by 3 and 5. It does not divide evenly by 2, 6, 8, 9, or 10.

Check: The 6 check needs both 2 and 3. The last-three-digit check for 8 uses 035, or 35.

Common mistakes and how to avoid them

Using the last digit to check for 4 or 8.

Correction: Use the last two digits for 4 and the last three digits for 8.

Thinking a number passes the rule for 6 because it passes the rule for 2 or for 3.

Correction: It must pass both the 2 and 3 checks.

Treating a digit sum that is not a multiple of 9 as a failure of the rule for 3.

Correction: Check the chosen divisor. A digit sum can pass the rule for 3 without passing the rule for 9.

Lesson summary

Dividing evenly means there is no remainder.
Use the last digit for 2, 5, and 10; the last two digits for 4; and the last three digits for 8.
Add the digits to check for 3 or 9.
For 6, the number must pass both the 2 and 3 rules.

Check your understanding

Question 1

Does 4,572 divide evenly by 3?

Yes, because its digit sum is 18.
No, because its last digit is 2.
No, because its last two digits are 72.
correctIndex

Show answer and explanation

Yes, because its digit sum is 18.

The digit sum is 18, and 18 divides evenly by 3.

Question 2

Does 1,236 divide evenly by 4?

No, because it ends in 6.
Yes, because its last two digits, 36, divide evenly by 4.
No, because its digit sum is 12.
correctIndex

Show answer and explanation

Yes, because its last two digits, 36, divide evenly by 4.

The rule for 4 checks the last two digits. 36 divides evenly by 4.

Question 3

Which statement about 2,418 is correct?

It divides evenly by 6 because it passes the rules for both 2 and 3.
It does not divide evenly by 2 because its last digit is 8.
It divides evenly by 9 because its digit sum is 15.
correctIndex

Show answer and explanation

It divides evenly by 6 because it passes the rules for both 2 and 3.

Its last digit is even, so it passes the 2 check. Its digit sum is 15, which passes the 3 check. It therefore passes the rule for 6.

Question 4

Does 3,152 divide evenly by 8?

Yes, because the last digit is 2.
No, because the last three digits, 152, do not divide evenly by 8.
Yes, because its digit sum is 11.
correctIndex

Show answer and explanation

No, because the last three digits, 152, do not divide evenly by 8.

The last three digits are 152. Since 152 is not divisible evenly by 8, the full number does not pass the rule for 8.

Key terms

Divides evenly: Dividing gives a whole-number answer with no remainder.
Divisor: The number you divide by.
Digit sum: The total when you add all the digits in a number.
Remainder: An amount left after dividing when the groups cannot be equal with none left over.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation B2.2. It is a study resource, not an official curriculum publication.

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