Prime Factorization

Prime Factorization

Writing an integer as a product of powers of prime numbers.

Table showing prime factorizations: 30=2·3·5, 24=2³·3, 1000=2³·5³, 2001=3·23·29

See also

Key Formula

n = p_1^{a_1} \times p_2^{a_2} \times p_3^{a_3} \times \cdots \times p_k^{a_k}

Where:

$n$ = The positive integer being factored (must be greater than 1)
$p_1, p_2, \ldots, p_k$ = Distinct prime numbers, listed in increasing order
$a_1, a_2, \ldots, a_k$ = Positive integer exponents indicating how many times each prime appears

Worked Example

Problem: Find the prime factorization of 360.

Step 1: Divide by the smallest prime, 2. Since 360 is even, divide repeatedly by 2.

360 \div 2 = 180,\quad 180 \div 2 = 90,\quad 90 \div 2 = 45

Step 2: 45 is odd, so 2 no longer divides evenly. Move to the next prime, 3.

45 \div 3 = 15,\quad 15 \div 3 = 5

Step 3: 5 is itself a prime number, so the division process stops here.

5 \div 5 = 1

Step 4: Collect all the prime factors and write them using exponents.

360 = 2^3 \times 3^2 \times 5^1

Answer: The prime factorization of 360 is

2^3 \times 3^2 \times 5

Another Example

This example uses a factor tree method instead of repeated division, and it features a larger number that turns out to be a perfect square — showing how prime factorization reveals structural properties of a number.

Problem: Find the prime factorization of 1,764.

Step 1: Build a factor tree. Start by splitting 1,764 into any two factors you recognize.

1{,}764 = 2 \times 882

Step 2: Continue splitting each composite number. 882 is even, so divide by 2 again.

882 = 2 \times 441

Step 3: 441 is not even. Check divisibility by 3: the digit sum is 4 + 4 + 1 = 9, which is divisible by 3.

441 = 3 \times 147,\quad 147 = 3 \times 49

Step 4: 49 is a perfect square of the prime 7.

49 = 7 \times 7

Step 5: All branches of the factor tree now end in primes. Combine them with exponents.

1{,}764 = 2^2 \times 3^2 \times 7^2

Answer: The prime factorization of 1,764 is

2^2 \times 3^2 \times 7^2

. Notice this equals

(2 \times 3 \times 7)^2 = 42^2

, confirming that 1,764 is a perfect square.

Frequently Asked Questions

What is the difference between factors and prime factorization?

Factors of a number are all the integers that divide it evenly. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Prime factorization specifically breaks 12 down into only prime factors:

2^2 \times 3

. Every factor of 12 can be built by choosing different combinations of these primes.

How do you find the prime factorization of a number quickly?

Start by dividing by the smallest prime (2) as many times as possible, then move to 3, then 5, then 7, and so on. You only need to test primes up to the square root of the remaining number. If the remaining number is greater than 1 at that point, it is itself prime. Divisibility rules for 2, 3, and 5 speed up the process significantly.

Why is 1 not included in prime factorization?

The number 1 is neither prime nor composite. Including 1 as a prime factor would violate the Fundamental Theorem of Arithmetic, which guarantees that every integer greater than 1 has a unique prime factorization. If 1 were prime, you could write

6 = 2 \times 3 = 1 \times 2 \times 3 = 1 \times 1 \times 2 \times 3

, making the factorization non-unique.

Prime Factorization vs. Factor List (All Factors)

	Prime Factorization	Factor List (All Factors)
Definition	Expressing a number as a product of primes with exponents	Listing every integer that divides the number evenly
Example for 60	2² × 3 × 5	1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Uniqueness	Always exactly one prime factorization per number	One complete list, but the count of factors varies
When to use	Finding GCF, LCM, simplifying fractions, testing if a number is a perfect square	Listing divisors, checking divisibility, counting factor pairs

Why It Matters

Prime factorization is the backbone of many procedures you encounter in middle and high school math. You use it to find the greatest common factor (GCF) and least common multiple (LCM) when adding fractions with unlike denominators or simplifying ratios. It also appears in algebra when factoring polynomials, in number theory, and is the mathematical foundation of modern encryption systems like RSA that secure online transactions.

Common Mistakes

Mistake: Stopping too early by including a composite factor in the result.

Correction: Every factor in your final answer must be prime. For instance, writing 36 = 4 × 9 is not a prime factorization because 4 and 9 are composite. You must continue: 36 = 2² × 3². Always check that each base in your answer is prime.

Mistake: Forgetting to use exponents and instead listing repeated primes.

Correction: While writing 72 = 2 × 2 × 2 × 3 × 3 is technically correct, standard prime factorization form uses exponents: 72 = 2³ × 3². Exponent form is expected on most tests and is far easier to work with when computing GCF or LCM.

Related Terms

Prime Number — The building blocks used in prime factorization
Factor Tree — A visual method for finding prime factorization
Product — Prime factorization expresses a number as a product
Power — Exponents used to write repeated prime factors
Integers — The set of numbers that can be prime factored
Greatest Common Factor (GCF) — Found by comparing prime factorizations
Least Common Multiple (LCM) — Found by using highest powers from prime factorizations
Fundamental Theorem of Arithmetic — Guarantees uniqueness of prime factorization