A 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.
A factor divides the number with nothing left over, and factors always come in pairs. 6 splits 24 into equal rows; 5 leaves a remainder, so 5 is not a factor of 24.
Formal definition and notation
For whole numbersa and b with a ≠ 0, a is a factor (or divisor) of b if there is a whole number k with a × k = b; equivalently, b ÷ a leaves remainder 0. Then b is a multiple of a, and a and k form a factor pair of b. In algebra, an expression is a factor of another if it divides it evenly: x is a factor of 3x, and (x + 2) is a factor of x2 + 5x + 6.
Symbol
Read it as
Meaning
a ∣ b
a divides b
a is a factor of b, for example 6 ∣ 24
a × b
a times b
in 4 × 6 = 24, the numbers 4 and 6 are the factors and 24 is the product
Key facts
1 is a factor of every whole number, and every whole number is a factor of itself.
Factors come in pairs that multiply to the number: 24 has the pairs 1 × 24, 2 × 12, 3 × 8 and 4 × 6.
A number with exactly two factors is a prime number (7: 1 and 7). A number with more than two is a composite number (24 has eight).
A factor of a number is never larger than the number (for positive whole numbers), but a multiple is never smaller.
A perfect square has an odd number of factors, because one pair repeats: 36 has nine factors, with 6 paired with itself.
How to find every factor
How to find every factor
Test d = 1, 2, 3, … while d × d ≤ n; each d that divides n gives the pair d and n ÷ d
n
the whole number you are factoring
d
a trial divisor; it is a factor when n ÷ d has no remainder
n ÷ d
the partner in the factor pair, found for free
You can stop once d × d passes n, because every pair has one member at or below √n. For 36 you test 1 to 6 and get the pairs 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6.
Try it: list the factors
Interactive
Type a whole number or tap a sample. The checker lists every factor of the number and draws a factor tree that ends in primes. Compare 36 (a perfect square) with 48, and see what happens with 17 and 1.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Factor examples, step by step
Example 1: List all the factors
Warm-up
List all the factors of 36.
Test 1, 2, 3, … and record each pair: 1 × 36, 2 × 18, 3 × 12, 4 × 9.
5 does not divide 36 (36 ÷ 5 = 7 R 1), so skip it.
6 × 6 = 36: 6 pairs with itself. Since 6 × 6 reaches 36, you can stop.
Write the factors in order, listing 6 once.
Answer: The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
Example 2: Test whether a number is a factor
Core
Is 7 a factor of 91? Is 8 a factor of 92?
Divide 91 by 7: 91 ÷ 7 = 13, with no remainder.
So 7 × 13 = 91, and 7 is a factor of 91.
Divide 92 by 8: 8 × 11 = 88, and 92 − 88 = 4, so 92 ÷ 8 = 11 R 4.
There is a remainder, so 8 is not a factor of 92.
Answer: Yes, 7 is a factor of 91. No, 8 is not a factor of 92.
Example 3: Equal rows of chairs
Test-style
A teacher wants to set up 48 chairs in equal rows with no chairs left over. Each row must have at least 4 and at most 10 chairs. How many different row lengths are possible?
Equal rows with nothing left over means the row length must be a factor of 48.
Factors of 12 divide 12: 1, 2, 3, 4, 6, 12. Multiples of 12 are 12 times a whole number: 12, 24, 36, and so on. A number has finitely many factors but infinitely many multiples.
Factorvs.Prime factor
A prime factor is a factor that is also prime. The factors of 12 are 1, 2, 3, 4, 6 and 12, but the prime factors are only 2 and 3.
A factor is one of the numbers or expressions that multiply together. Factoring is the process of rewriting something as a product of factors, such as 18 = 2 × 3 × 3.
Quick check: 3 questions
Question 1 of 3
Which number is NOT a factor of 30?
Show answer and explanation
Answer: C.30 ÷ 4 = 7 R 2, so 4 is not a factor. The others divide evenly: 5 × 6 = 30 and 2 × 15 = 30. Being even is not enough: 4 needs 30 to be a multiple of 4.
Question 2 of 3
How many factors does 16 have?
Show answer and explanation
Answer: B. The factors are 1, 2, 4, 8 and 16: five in all, because 4 pairs with itself. Choice A forgets 16 or 1, and 8 is half of 16, not the number of factors.
Question 3 of 3
What is the greatest factor of 45 that is less than 45?
Show answer and explanation
Answer: A. The smallest factor above 1 is 3, so the largest factor below 45 is its partner: 45 ÷ 3 = 15. 22.5 is half of 45 but not a whole number, and 9 and 5 are factors, but smaller ones.
Factor: frequently asked questions
Is a factor the same as a divisor?
For whole numbers, yes. A divisor of 20 is any number that divides 20 evenly, which is exactly what a factor is: 1, 2, 4, 5, 10 and 20. Some textbooks use divisor when they divide and factor when they multiply, but the list of numbers is the same. Spreadsheets and calculators often call them divisors.
Is 1 a factor of every number?
Yes. Any whole number n equals 1 × n, so 1 and n are always a factor pair. That is why every whole number greater than 1 has at least two factors. Numbers with only those two are prime, like 13. The number 1 itself has just one factor, and 0 has every nonzero whole number as a factor.
How do you know when you have found all the factors?
Test divisors in order, 1, 2, 3 and so on, writing each factor pair as you go. Stop when the divisor times itself is larger than the number. For 60 you test 1 through 7 and get the pairs 1 and 60, 2 and 30, 3 and 20, 4 and 15, 5 and 12, 6 and 10: twelve factors.
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 5DivisorA 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 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 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 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 2Prime 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 6Composite NumberA composite number is a whole number greater than 1 that has more than two factors. It can be divided evenly by a number other than 1 and itself: 12 is composite because 1, 2, 3, 4, 6 and 12 all divide it. Every whole number above 1 is either prime or composite; 1 is neither.Number System · Grade 4, Grade 5, Grade 6Perfect SquareA perfect square is a number that equals a whole number times itself, so its square root is a whole number. For example, 49 is a perfect square because 7 × 7 = 49 and √49 = 7. The first perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.Number System · Grade 6, Grade 7, Grade 8Odd NumberAn odd number is a whole number that cannot be split into two equal groups: dividing it by 2 always leaves a remainder of 1. Odd numbers end in 1, 3, 5, 7 or 9, such as 7, 23 and 409. Every odd number can be written as 2k + 1, one more than a multiple of 2.Number System · Grade 2, Grade 3, Grade 4LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7, Grade 8ProductThe product is the answer you get when you multiply two or more numbers. In 4 × 6 = 24, the product is 24, and 4 and 6 are the factors. In word problems, the words “product of” always mean multiply, so “the product of 3 and 7” is 21.Operations · Grade 3, Grade 4, Grade 5MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4Greatest 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 7Least Common MultipleThe least common multiple (LCM) is the smallest positive multiple shared by two or more numbers. The LCM of 6 and 9 is 18: it is the first number on both lists of multiples, 6, 12, 18, … and 9, 18, …. The LCM of the denominators is the least common denominator for adding fractions.Number System · Grade 5, Grade 6, Grade 7Prime 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 6Divisibility RulesDivisibility 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.Number System · Grade 4, Grade 5, Grade 6