A 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.
In Spanish
par de factores
Also called
factor pairs, pair of factors
Grade level
K-5
Grade 4Grade 5Grade 6CCSS 4.OA.B.4TABE
Factor Pair in a picture
Each arc joins a factor pair of 36. The pairs meet in the middle at 6, the square root of 36, which is where you can stop searching.
Formal definition and notation
For a whole numbern ≥ 1, a factor pair of n is an unordered pair of whole numbers {a, b} with a × b = n. If a ≤ b, then a ≤ √n, so every factor pair is found by testing a = 1, 2, …, ⌊√n⌋. A prime has exactly one factor pair, 1 × p; a perfect square has one pair with equal numbers. With integers, (−a) × (−b) = n gives negative factor pairs too.
Symbol
Read it as
Meaning
a × b = n
a times b equals n
a and b form a factor pair of n
Key facts
Every whole number greater than 0 has the factor pair 1 and itself.
Each factor pair matches one rectangular array: the pairs of 12 give 1 by 12, 2 by 6 and 3 by 4 arrays.
Pairs come in twos of factors, so a number has an even count of factors unless it is a perfect square, where one pair repeats a number.
Order does not make a new pair: 3 × 4 and 4 × 3 are the same factor pair.
Factor pairs drive factoring in algebra: to factor x2 + 7x + 12, find the pair of 12 that adds to 7, which is 3 and 4.
Finding all factor pairs
Finding all factor pairs
Test a = 1, 2, 3, … while a × a ≤ n
a
the smaller number of the pair
n ÷ a
its partner, when the division has no remainder
For 60: 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, 6 × 10. Since 8 × 8 > 60, stop after 7. That is 6 pairs.
Try it: factor pair finder
Interactive
Type a number to see all of its factor pairs. Compare 48 with 49 and 47: a perfect square has a pair with equal numbers, and a prime has only one pair.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
A factor is one number that divides evenly; a factor pair is two factors that multiply to the number. 3 is a factor of 18; 3 and 6 are a factor pair.
Factor Pairvs.Prime factorization
Prime factorization breaks a number all the way into primes: 36 = 22 × 32. A factor pair stops at two numbers, like 4 × 9.
Quick check: 3 questions
Question 1 of 3
Which is NOT a factor pair of 30?
Show answer and explanation
Answer: D.4 × 7 = 28, not 30. The others multiply to 30: 2 × 15, 5 × 6, 3 × 10.
Question 2 of 3
How many factor pairs does 64 have?
Show answer and explanation
Answer: B.1 × 64, 2 × 32, 4 × 16, 8 × 8: 4 pairs. 7 counts the factors instead of the pairs, 3 forgets 8 × 8, and 6 counts reversed pairs.
Question 3 of 3
Which number has exactly one factor pair?
Show answer and explanation
Answer: A. 31 is prime, so its only pair is 1 × 31. 33 = 3 × 11, 35 = 5 × 7 and 39 = 3 × 13 have two pairs each.
Factor Pair: frequently asked questions
How do you find all the factor pairs of a number?
Start with 1 and the number. Then try 2, 3, 4 and so on: each time a number divides evenly, write it with its quotient as a pair. Stop when the number you are testing times itself is bigger than the target. For 45: 1 × 45, 3 × 15, 5 × 9; since 7 × 7 = 49 > 45, you are done.
Can factor pairs be negative?
Yes, if you work with integers. Because a negative times a negative is positive, −3 × −4 = 12, so −3 and −4 are a factor pair of 12. In elementary grades factor pairs mean whole numbers only, but algebra uses negative pairs to factor expressions such as x2 − 7x + 12 = (x − 3)(x − 4).
FactorA 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 6Whole 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 5Perfect 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 8NumberA 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 2ArrayAn array is an arrangement of objects in equal rows and equal columns, forming a rectangle. A 4 by 6 array has 4 rows of 6, so it holds 4 × 6 = 24 objects. Arrays show multiplication as equal groups, and turning one sideways shows that 4 × 6 = 6 × 4.Operations · Grade 2, Grade 3, Grade 4Prime 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 6FactoringFactoring 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 8RemainderA 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 6SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3MeterA meter (m) is a metric unit of length equal to 100 centimeters. It is the base unit of length in the metric system, a little over three feet (about 3.28 feet). A doorway is about 2 meters tall, and 1,000 meters make a kilometer. To change meters to centimeters, multiply by 100.Measurement · Grade 3, Grade 4, Grade 5SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2PerimeterPerimeter is the total distance around a two-dimensional shape. To find it, add the lengths of all the sides. A rectangle with sides 5 and 3 has perimeter 5 + 3 + 5 + 3 = 16. Perimeter is a length, so it uses length units such as inches or meters, never square units.Geometry · Grade 3, Grade 4, Grade 5Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1Prime 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 6MultiplicationMultiplication 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 7MultipleA 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 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