Divisibility rules are shortcuts that tell whether one whole number divides another with no remainder, without doing the division. A number is divisible by 2 if its last digit is even, by 3 if its digit sum is divisible by 3, by 5 if it ends in 0 or 5, and by 10 if it ends in 0.
In Spanish
reglas de divisibilidad
Also called
divisibility tests, rules of divisibility, divisibility tricks
Each divisibility rule checks a small piece of the number instead of dividing the whole thing.
Formal definition and notation
A whole numbern is divisible by d ≠ 0 when n = d × k for some whole number k, written d ∣ n. Divisibility rules follow from place value: since 10 leaves remainder 1 when divided by 3 or 9, n and its digit sum leave the same remainder, and since 100 is divisible by 4 and 1,000 by 8, only the last two or three digits decide divisibility by 4 or 8.
Symbol
Read it as
Meaning
d ∣ n
d divides n
n is divisible by d: 3 ∣ 471
Key facts
By 2: last digit 0, 2, 4, 6 or 8. By 5: last digit 0 or 5. By 10: last digit 0.
By 3: the digit sum is divisible by 3. By 9: the digit sum is divisible by 9. Example: 5,832 has digit sum 18, so it is divisible by both.
By 4: the last two digits form a number divisible by 4. By 8: the last three digits do.
By 6: divisible by both 2 and 3. Combine rules for other products of different primes, like 15 = 3 × 5.
By 11: alternately subtract and add the digits; if the result is divisible by 11, so is the number. For 2,728: 2 − 7 + 2 − 8 = −11.
By 7: double the last digit and subtract it from the rest; repeat. For 343: 34 − 6 = 28, and 28 is divisible by 7.
Try it: check your divisibility test
Interactive
Apply a rule in your head, then type the number to see its full list of factors. Try 2,736 (digit sum 18) and 1,001 (divisible by 7, 11 and 13).
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Divisibility Rules examples, step by step
Example 1: Test 2,345
Warm-up
Is 2,345 divisible by 2, 3, 5 or 10?
Last digit 5 is odd: not divisible by 2 or 10.
Last digit 5: divisible by 5.
Digit sum 2 + 3 + 4 + 5 = 14, not a multiple of 3: not divisible by 3.
Answer: 2,345 is divisible by 5 only (of 2, 3, 5 and 10).
Example 2: Find the missing digit
Core
What digit can replace the box in 4,7□ 2 so the number is divisible by 9?
Known digit sum: 4 + 7 + 2 = 13.
The full sum must be a multiple of 9: 18 is the only one possible, since the box is 0 to 9.
Box = 18 − 13 = 5. Check: 4,752 ÷ 9 = 528.
Answer: The digit is 5, making 4,752.
Example 3: Divisible by 12?
Test-style
Is 9,516 divisible by 12?
12 = 3 × 4, and 3 and 4 share no factor, so test both.
By 4: last two digits 16, and 16 ÷ 4 = 4. Yes.
By 3: 9 + 5 + 1 + 6 = 21, a multiple of 3. Yes.
So 9,516 is divisible by 12: 9,516 ÷ 12 = 793.
Answer: Yes, 9,516 is divisible by 12.
Common mistakes
✗ Mistake316 is divisible by 3 because it ends in 6.
✓ InsteadDigit sum 3 + 1 + 6 = 10, not a multiple of 3, so 316 is not divisible by 3.
The rule for 3 uses the digit sum, not the last digit.
✗ Mistake1,324 is divisible by 8 because 24 is divisible by 8.
✓ InsteadUse the last three digits: 324 ÷ 8 = 40.5, so 1,324 is not divisible by 8.
The test for 4 uses two digits; the test for 8 needs three.
✗ Mistake24 is divisible by 2 and 4, so every number divisible by 2 and 4 is divisible by 8.
✓ Instead12 is divisible by 2 and 4 but not by 8.
Combining rules only works for factors that share no common factor, like 3 and 4 for 12.
If a number is divisible by 6, then 6 is a factor of it. Divisibility rules are a fast way to find factors.
Quick check: 3 questions
Question 1 of 3
Which number is divisible by 9?
Show answer and explanation
Answer: A. 4,518 has digit sum 4 + 5 + 1 + 8 = 18, a multiple of 9. The others have digit sums 19, 10 and 15; 6,135 is divisible by 3 (15) but not by 9.
Question 2 of 3
Which number is divisible by 6?
Show answer and explanation
Answer: C. 5,412 is even and its digit sum 5 + 4 + 1 + 2 = 12 is a multiple of 3. 3,124 is even but its digit sum is 10; 2,715 and 4,503 are odd.
Question 3 of 3
A number ends in 30. Which must be true?
Show answer and explanation
Answer: D. Any number ending in 0 is divisible by 10 (and 2 and 5). It need not be divisible by 4 (130 is not), 3 (130 is not) or 9.
Divisibility Rules: frequently asked questions
Why does the divisibility rule for 3 work?
Each place value is 1 more than a multiple of 3: 10 = 9 + 1, 100 = 99 + 1, 1,000 = 999 + 1. So 471 is 4 × 99 + 7 × 9, which is a multiple of 3, plus 4 + 7 + 1. The number is divisible by 3 exactly when that digit sum is. The same reasoning gives the rule for 9.
Is there a divisibility rule for 7?
Yes, though it is slower than dividing for small numbers. Double the last digit and subtract it from the number formed by the other digits; repeat until the result is small. If the result is divisible by 7, so is the original. For 637: 63 − 14 = 49, and 49 = 7 × 7, so 637 is divisible by 7.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 4.OA.B.4
Cite this entry
Effortless Math Team. “Divisibility Rules.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/divisibility-rules/
Flashcard
TermDivisibility Rulesby 3? add the digits
DefinitionQuick tests for dividing evenly. Ends in 0 or 5: divisible by 5. Digit sum a multiple of 3: divisible by 3, as with 471.
Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5Place ValuePlace value is the value of a digit based on its position in a number. In 3,507 the 5 is worth 500, because it sits in the hundreds place. Each place is worth 10 times the place to its right: ones, tens, hundreds, thousands, and after the decimal point, tenths and hundredths.Number System · Grade 3, Grade 4, Grade 5RemainderA remainder is the amount left over when one whole number does not divide another evenly. In 23 ÷ 5, four groups of 5 make 20 and 3 are left, so 23 ÷ 5 = 4 R 3. The remainder is always smaller than the divisor. A remainder of 0 means the division is exact.Operations · Grade 4, Grade 5, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2DivisorA divisor is the number you divide by. In 56 ÷ 8 = 7, the divisor is 8: it tells how many equal groups to make, or how big each group is. The divisor can never be 0. A divisor of a whole number is also one of its factors, because it divides that number with no remainder.Operations · Grade 3, Grade 4, Grade 5FactorA factor is a number or expression that divides another number or expression evenly, with no remainder. Factors multiply together to make a product: 4 × 6 = 24, so 4 and 6 are factors of 24. Every whole number greater than 1 has at least two factors, 1 and itself.Number System · Grade 4, Grade 5, Grade 6DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5DigitA digit is one of the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 that we use to write every number. The number 507 has three digits: 5, 0 and 7. What a digit is worth depends on its place: in 507, the 5 stands for 5 hundreds, or 500.Number System · Grade 1, Grade 2, Grade 3Prime NumberA prime number is a whole number greater than 1 with exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13. The number 1 is not prime, and 2 is the only even prime number. Whole numbers greater than 1 that are not prime are composite.Number System · Grade 4, Grade 5, Grade 6Prime FactorizationPrime factorization is writing a whole number as a product of prime numbers only, such as 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32. Find it with a factor tree or repeated division. Every whole number greater than 1 has exactly one prime factorization, apart from the order of the factors.Number System · Grade 4, Grade 5, Grade 6Greatest Common FactorThe greatest common factor (GCF) is the largest factor shared by two or more numbers: the largest number that divides into both without a remainder. The GCF of 18 and 24 is 6. Find it by listing factors or by multiplying the prime factors the numbers share. The GCF is how you simplify fractions in one step.Number System · Grade 5, Grade 6, Grade 7Simplest FormSimplest form is the way of writing a fraction so its numerator and denominator have no common factor other than 1. To get there, divide the top and bottom by their greatest common factor: 68 = 6 ÷ 28 ÷ 2 = 34. The value stays the same; only the numbers get smaller. It is also called lowest terms.Number System · Grade 4, Grade 5, Grade 6MultipleA multiple of a number is the result of multiplying that number by any whole number. The multiples of 4 are 4, 8, 12, 16, and so on, continuing without end. Every number is a multiple of itself and of 1. To test a number, divide: if the remainder is 0, it is a multiple.Number System · Grade 4, Grade 5, Grade 6Factor PairA factor pair is two whole numbers that multiply to give a certain number. The factor pairs of 12 are 1 × 12, 2 × 6 and 3 × 4. Listing factor pairs is a sure way to find every factor of a number: start at 1 and stop once the pairs begin to repeat.Number System · Grade 4, Grade 5, Grade 6