Prime 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.
In Spanish
factorización prima
Also called
prime decomposition, prime factors, factor tree, product of primes
Start a factor tree with any factor pair. However you split, the branches end in the same primes, which is why each number has just one prime factorization.
Formal definition and notation
The prime factorization of a whole numbern ≥ 2 is its expression as a product n = p1a1 × p2a2 × ⋯ × pkak, where the pi are distinct prime numbers and the exponentsai are positive whole numbers. By the Fundamental Theorem of Arithmetic, this product is unique apart from the order of the factors. A prime number is its own prime factorization, and 1 has none.
Symbol
Read it as
Meaning
2 × 2 × 2 × 3 × 3
two times two times two times three times three
the prime factorization written out in full
23 × 32
two cubed times three squared
the same factorization in exponent form
pa
p to the a
the prime p appears a times
Key facts
Only prime numbers may appear: 2, 3, 5, 7, 11, 13, … Never write 1 or a composite number in the final answer.
The prime factorization of a number is unique: every factor tree of 72 ends with three 2s and two 3s.
To test for primes when dividing, you only need to try primes up to the square root of the number.
The GCF of two numbers multiplies the primes they share (lowest powers); the LCM multiplies every prime that appears (highest powers).
A number is a perfect square exactly when every exponent in its prime factorization is even: 36 = 22 × 32.
Two ways to find it
Two ways to find it
factor tree: split into any factor pair, repeat until every branch ends in a prime
divide by the smallest prime that works, again and again, until the quotient is 1
Both methods give the same primes. Write them from least to greatest and group repeats with exponents.
Try it: grow a factor tree
Interactive
Type a whole number or pick a sample. The tree splits it until every branch ends in a prime, then writes the prime factorization with exponents. Try 97 to see a prime with no branches.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Prime Factorization examples, step by step
Example 1: A factor tree
Warm-up
Find the prime factorization of 90.
Split 90 into 9 × 10.
9 = 3 × 3 and 10 = 2 × 5.
Every branch is prime: 2, 3, 3, 5.
Answer:90 = 2 × 32 × 5.
Example 2: The ladder method
Core
Find the prime factorization of 588.
588 ÷ 2 = 294, 294 ÷ 2 = 147.
147 is odd. 147 ÷ 3 = 49.
49 ÷ 7 = 7, and 7 ÷ 7 = 1.
The primes used: 2, 2, 3, 7, 7.
Answer:588 = 22 × 3 × 72.
Example 3: GCF and LCM from prime factorizations
Test-style
Use prime factorization to find the GCF and LCM of 84 and 90.
A prime number has exactly two factors. Prime factorization breaks any whole number into those primes.
Quick check: 3 questions
Question 1 of 3
What is the prime factorization of 60?
Show answer and explanation
Answer: C.60 = 2 × 2 × 3 × 5 = 22 × 3 × 5. A and B stop at composites (30, 4), and D drops a 2, which gives 30.
Question 2 of 3
Which number has the prime factorization 23 × 5?
Show answer and explanation
Answer: B.23 × 5 = 8 × 5 = 40. 30 reads 23 as 2 × 3, 16 is 24, and 80 is 24 × 5.
Question 3 of 3
Which is NOT a correct prime factorization?
Show answer and explanation
Answer: D. 6 is not prime, so 6 × 7 is a factorization but not a prime one; the prime factorization of 42 is 2 × 3 × 7. The other three use only primes and multiply correctly.
Prime Factorization: frequently asked questions
Can a number have more than one prime factorization?
No. Apart from the order of the factors, every whole number greater than 1 has exactly one prime factorization. This fact is called the Fundamental Theorem of Arithmetic. Two people can start a factor tree with different splits, such as 72 = 8 × 9 and 72 = 2 × 36, and they will still end with the same primes.
What is the prime factorization of a prime number?
The prime number itself. For example, the prime factorization of 13 is just 13, because 13 has no factors except 1 and itself, so a factor tree cannot split it. The number 1 is a special case: it is neither prime nor composite and has no prime factorization.
Why is prime factorization useful?
It makes many fraction and number problems easier. You can read off the greatest common factor and least common multiple, simplify fractions by canceling shared primes, find common denominators, test whether a number is a perfect square or cube, and simplify square roots: √72 = √22 × 32 × 2 = 6√2.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 4.OA.B.4, CCSS 6.NS.B.4
Cite this entry
Effortless Math Team. “Prime Factorization.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/prime-factorization/
Flashcard
TermPrime Factorizationa product of primes only
DefinitionWriting a number as a product of primes: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3². Every number above 1 has exactly one.
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 5Prime 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 6ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8Composite 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 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 2Square 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 1PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7Perfect 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 8FactorA 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 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 8PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2Greatest 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 7SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6Common DenominatorA common denominator is a number that can be used as the denominator of two or more fractions at once. It is any common multiple of the denominators. For 13 and 14, 12 works: 412 and 312. You need one to add, subtract or compare fractions with different denominators.Number System · Grade 4, Grade 5, Grade 6Simplest 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 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 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