“Prime factorize 360” - can you still do it? Prime factorization is a core middle-school skill that is easy to forget.

This article explains how to do prime factorization with worked examples: what primes are, dividing by small primes step by step, factor trees, and counting divisors.

What is a prime number?

A prime number is an integer greater than 1 that is divisible only by 1 and itself. In order, they begin:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
Number Prime? Reason
1 No Its only divisor is itself, so it fails the definition
2 Yes Divisible only by 1 and 2 (the only even prime)
4 No Divisible by 2
9 No Divisible by 3

Integers greater than 1 that are not prime are called composite numbers.

What is prime factorization?

Prime factorization means writing an integer as a product of primes only. For example, 360 becomes:

360 = 2 × 2 × 2 × 3 × 3 × 5

Grouping equal primes gives the exponent form:

360 = 2³ × 3² × 5

Apart from the order of factors, this representation is unique - the fundamental theorem of arithmetic. Each prime is called a prime factor.

How to factorize: divide by small primes

The method is simple: divide by the smallest primes in order.

  1. Divide by 2 as many times as possible
  2. Move on to 3, 5, 7, 11…, trying each in turn
  3. Stop when the quotient is itself prime
  4. Collect the primes you divided by as a product

Example 1: factorizing 360

360 ÷ 2 = 180
180 ÷ 2 = 90
90 ÷ 2 = 45   (2 used three times)
45 ÷ 3 = 15
15 ÷ 3 = 5     (3 used twice)
5 is prime, so stop

Collecting the divisors gives 360 = 2 × 2 × 2 × 3 × 3 × 5, or 2³ × 3² × 5.

Example 2: factorizing 990

990 ÷ 2 = 495   (2 once)
495 ÷ 3 = 165
165 ÷ 3 = 55    (3 twice)
55 ÷ 5 = 11     (5 once)
11 is prime, so stop

The answer is 990 = 2 × 3² × 5 × 11.

Example 3: factorizing 504

504 ÷ 2 = 252
252 ÷ 2 = 126
126 ÷ 2 = 63   (2 three times)
63 ÷ 3 = 21
21 ÷ 3 = 7      (3 twice)
7 is prime, so stop

So 504 = 2³ × 3² × 7, and the divisor count is (3+1) × (2+1) × (1+1) = 24.

How far should you keep testing?

You only need to test divisors up to the square root of the current quotient. For 97, since 10 × 10 > 97, trying 2, 3, 5 and 7 is enough to conclude it is prime.

Drawing a factor tree

A factor tree shows the same process visually. Split 360 into 36 × 10, split 36 into 6 × 6 and 10 into 2 × 5, and keep splitting until every branch ends in a prime.

360
├─ 36 ─┬─ 6 ─┬─ 2
│      │     └─ 3
│      └─ 6 ─┬─ 2
│            └─ 3
└─ 10 ─┬─ 2
       └─ 5

Multiplying the primes at the tips gives the same answer. Either method works; dividing by small primes in order is more mistake-proof.

Common mistakes: counting 1 as a prime, forgetting to include the final quotient when it is prime, and dividing by composite numbers such as 4 or 9. Always divide by primes, and check whether the last quotient is prime.

Counting divisors

Once you have the factorization, counting divisors is quick: add 1 to each exponent and multiply.

For 360 = 2³ × 3² × 5:

  • exponent 3 → (3 + 1) = 4
  • exponent 2 → (2 + 1) = 3
  • exponent 1 → (1 + 1) = 2
  • divisors = 4 × 3 × 2 = 24

Why? You can use 2 zero to three times (4 choices), 3 zero to two times (3 choices) and 5 zero or once (2 choices), and every combination gives a distinct divisor.

Practice: divisors of 180

180 = 2² × 3² × 5, so the divisor count is (2 + 1) × (2 + 1) × (1 + 1) = 18.

Where it pays off: prime factorization is the groundwork for greatest common divisors and least common multiples. Line up the factors of two numbers: the shared primes give the GCD, and the combined primes give the LCM.

Uses of prime factorization

Factorization is a building block for other problems:

  • Greatest common divisor: multiply the shared primes (24 = 2³×3 and 36 = 2²×3² share 2²×3 = 12)
  • Least common multiple: take every prime with the largest exponent (24 and 36 give 2³×3² = 72)
  • Divisor count: the exponent-plus-one product above

Fluency with factorization makes divisor and multiple problems far easier.

How to factorize with the tool

The Tools Hub prime factorization tool shows the factors and divisor count the moment you enter a number.

1

Enter an integer

Type an integer of 2 or greater (up to 15 digits).

2

Read the product and exponent forms

See both "2×2×2×3×3×5" and "2³×3²×5" styles at once.

3

Check the divisor count

The number of divisors and a prime check are shown too - handy for homework.

The tool introduced in this article

Prime Factorization Tool

Product form, exponent form, divisor count and a prime check - all in your browser.

Factorize now

Summary

  • A prime is an integer greater than 1 divisible only by 1 and itself (1 is not prime)
  • Prime factorization writes an integer as a product of primes, uniquely apart from order
  • The method: divide by the smallest primes in order, stop when the quotient is prime
  • The exponent form (2³×3²×5) is the compact way to write the result
  • Count divisors by adding 1 to each exponent and multiplying
  • The Tools Hub tool shows both forms and the divisor count instantly

Next time you need to factorize, follow the steps above or use the Tools Hub prime factorization tool.

FAQ

Why is 1 not a prime number?

A prime must have exactly two divisors: 1 and itself. The number 1 has only one divisor, so it does not meet the definition.

Is the prime factorization unique?

Yes. Apart from the order of the factors, every integer greater than 1 has exactly one prime factorization - the fundamental theorem of arithmetic.

How do I count the divisors?

Add 1 to each exponent and multiply the results. For 360 = 2³×3²×5, that is (3+1) × (2+1) × (1+1) = 24 divisors.

Can I factorize very large numbers?

The Tools Hub tool handles integers up to 15 digits. Larger numbers take noticeably longer, but 15 digits comfortably covers school-level problems.