Divisibility rules are shortcuts for deciding whether one whole number divides another with no remainder. They help you factor numbers faster, simplify fractions, check arithmetic, and recognize patterns in place value. Instead of doing long division every time, you can inspect digits, digit sums, or the last few digits.
These rules are especially useful when working with factors, multiples, prime factorization, and mental math.
Key Facts
- Divisible by 2: the last digit is even, so it is 0, 2, 4, 6, or 8.
- Divisible by 3: the sum of the digits is divisible by 3. Divisible by 9: the sum of the digits is divisible by 9.
- Divisible by 4: the last two digits form a number divisible by 4. Divisible by 8: the last three digits form a number divisible by 8.
- Divisible by 5: the last digit is 0 or 5. Divisible by 10: the last digit is 0.
- Divisible by 6: the number must be divisible by both 2 and 3, since 6 = 2 × 3.
- Divisible by 11: the alternating sum of the digits is divisible by 11, including 0. Example: for 583, 5 - 8 + 3 = 0, so 583 is divisible by 11.
Vocabulary
- Divisible
- A number is divisible by another number if division gives a whole number with no remainder.
- Factor
- A factor is a whole number that divides another whole number exactly.
- Multiple
- A multiple is the product of a number and a whole number.
- Remainder
- A remainder is the amount left over after division when the divisor does not divide the number exactly.
- Digit sum
- A digit sum is the sum of all the digits in a number, often used to test divisibility by 3 or 9.
Common Mistakes to Avoid
- Using the last digit rule for 4 or 8 is wrong because those tests depend on the last two digits for 4 and the last three digits for 8.
- Thinking a number divisible by 3 is automatically divisible by 9 is wrong because divisibility by 9 requires the digit sum to be a multiple of 9.
- Testing divisibility by 6 using only divisibility by 3 is wrong because the number must also be even.
- Adding the digits for the 11 rule is wrong because the rule uses an alternating sum, such as first digit minus second digit plus third digit.
Practice Questions
- 1 Determine which of these numbers are divisible by 2, 3, 4, 5, 6, 8, 9, 10, and 11: 360, 924, and 1001.
- 2 Use divisibility rules to factor 1,188 as much as possible without starting with long division.
- 3 Explain why the divisibility rule for 3 works using the idea that 10 = 9 + 1 and each place value leaves the same remainder as its digit when divided by 3.