Maths · Suggested Grade 6
Class 6 Divisibility Rules: 2, 3, 4, 5, 6, 8, 9, 10 & 11
Being even is enough for divisibility by 2. It is not enough for divisibility by 6. A good divisibility check names every condition that must hold.
Original explanations and practice by BananaLearn · Published 31 October 2026
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What divisible means
A whole number is divisible by a positive whole number if the division leaves remainder zero. For example, 348 is divisible by 6 because 348 = 6 × 58. The test predicts that exact division is possible; it does not necessarily tell you the quotient. We use positive divisors and whole-number examples in this guide.
You need place value, multiplication facts and the meaning of a remainder. Start by writing a number as hundreds, tens and ones. Each rule works because a large part of that place-value expansion is already divisible by the divisor, leaving a smaller part to check.
Use the last digit for 2, 5 and 10
For 2, the last digit must be 0, 2, 4, 6 or 8. Tens, hundreds and higher powers of ten are already even. For 5, the last digit must be 0 or 5. For 10, it must be 0. A number ending in 0 therefore passes all three tests, while one ending in 5 passes the test for 5 but not for 2 or 10.
Consider 735: its final digit gives yes for 5 and no for 2 or 10. Do not use the first digit or the total number of digits. A four-digit number is not automatically divisible by 4, and a large number can fail a test that a smaller one passes.
Use digit sums for 3 and 9
A number has the same remainder as its digit sum when divided by 3 or by 9. For 3, check whether the digit sum is a multiple of 3; for 9, check whether it is a multiple of 9. For 348 the sum is 3 + 4 + 8 = 15, so it passes the test for 3 and fails the test for 9.
One explanation uses 348 = 3 × 99 + 4 × 9 + 15. The first two parts are multiples of 9 and therefore also of 3. Only the 15 needs checking. A number divisible by 9 is always divisible by 3, but the converse is false: 348 is a counterexample.
Use the last two or three digits for 4 and 8
For 4, check the number formed by the last two digits, because 100 is divisible by 4. For 8, check the last three digits, because 1000 is divisible by 8. Thus 3,416 passes the test for 8 because 416 = 8 × 52. It also passes the test for 4 because 16 is divisible by 4.
Leading zeros in the ending do not change its value: the last three digits of 5,024 form 024, meaning 24. A common mistake is checking only the last digit for 4. The numbers 12 and 22 both end in 2, but only 12 is divisible by 4.
Combine conditions for 6, and test 11 carefully
For 6, the number must pass both the 2 and 3 tests. The word both matters. The number 14 is even but its digit sum is 5, so it is not divisible by 6. The number 123 passes the 3 test but is odd, so it also fails the 6 test.
For 11, find the difference between the sums of digits in alternating positions. If that difference is 0 or a multiple of 11, the number is divisible by 11. For 1,331, the alternating sums are 1 + 3 = 4 and 3 + 1 = 4, so the difference is 0. Either direction of subtraction is acceptable if you recognise positive or negative multiples of 11.
Worked example: find a missing digit
Which digit can replace □ in 4□2 to make a multiple of 9? The digit sum is 4 + □ + 2 = 6 + □. Since □ is from 0 to 9, that sum lies from 6 to 15. Its only multiple of 9 is 9, so □ = 3. Check: 432 ÷ 9 = 48.
For divisibility by 6, the same number 4□2 is already even. Its digit sum must be divisible by 3, giving □ = 0, 3, 6 or 9. There can be several answers unless the question adds another condition. Writing only the first digit that works does not establish that it is the only one.
Choose tests for a purpose, not a memory race
To decide whether 48 objects can be packed in equal groups of 6, test both 2 and 3 or use the known fact 6 × 8 = 48. The rule is a tool, not a requirement to avoid efficient multiplication facts. For factor questions, record the divisor, the part you checked and the conclusion.
Do not combine arbitrary divisibility tests without care. Being divisible by 2 and 4 does not prove divisibility by 8: 12 passes both smaller tests and fails the 8 test. The combination works for 6 because 2 and 3 share no factor greater than 1. Extend that reasoning only after the basic tests are secure.
A divisibility evidence table
Copy the record into a notebook or use it to organise your own answer. Blank spaces are for your work.
| Number | Divisor | Part of the number checked | Conclusion and reason |
|---|---|---|---|
| 348 | 6 | ___ | ___ |
| 3,416 | 8 | ___ | ___ |
| 1,331 | 11 | ___ | ___ |
| Your choice | ___ | ___ | ___ |
10 questions with explained answers
Try the questions before opening the answers. Explain the reason, then try a changed example to check your understanding.
Is 735 divisible by 3?
Show answer and explanation
Yes
Its digits sum to 15, a multiple of 3.Is 735 divisible by 9?
Show answer and explanation
No
15 is not a multiple of 9.Is 1,024 divisible by 8?
Show answer and explanation
Yes
The last three digits form 24, which is 8 × 3.Is 1,024 divisible by 4?
Show answer and explanation
Yes
Its last two digits form 24.Is 14 divisible by 6?
Show answer and explanation
No
It passes 2 but fails 3.Is 123 divisible by 6?
Show answer and explanation
No
It passes 3 but fails 2.Complete 4□2 to make a multiple of 9.
Show answer and explanation
432
6 + □ must equal 9, so the missing digit is 3.List all digits that make 4□2 divisible by 6.
Show answer and explanation
0, 3, 6, 9
It is already even; the digit sum must be a multiple of 3.Is 2,541 divisible by 11?
Show answer and explanation
Yes
Alternating sums 2 + 4 and 5 + 1 are both 6.Does divisibility by both 2 and 4 prove divisibility by 8?
Show answer and explanation
No
12 is divisible by 2 and 4 but not by 8.
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