Pythagorist
Numbers

Prime Factorisation Calculator

Break any number down into its prime factors using the factor tree (repeated division) method. Results shown in index form (e.g. 360 = 2³ × 3² × 5) with full step-by-step working.

360 =

2³ × 3² × 5

How it works

  1. Divide by 2

    360÷2=180360 \div 2 = 180

    360 is even, so divide by 2. Quotient: 180.

  2. Divide by 2

    180÷2=90180 \div 2 = 90

    180 is even, so divide by 2. Quotient: 90.

  3. Divide by 2

    90÷2=4590 \div 2 = 45

    90 is even, so divide by 2. Quotient: 45.

  4. Divide by 3

    45÷3=1545 \div 3 = 15

    45 is divisible by 3. Quotient: 15.

  5. Divide by 3

    15÷3=515 \div 3 = 5

    15 is divisible by 3. Quotient: 5.

  6. Remaining factor

    The remaining quotient 5 is greater than 1 and is itself a prime factor.

    Result:5 is prime

  7. Write as a product of prime factors

    360 = 2³ × 3² × 5

    360 expressed as a product of its prime factors is 2³ × 3² × 5.

    Result:2³ × 3² × 5

The formula

n=p1a1×p2a2×⋯×pkakn = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}
p1,p2...p₁, p₂ ...Prime factors (in ascending order)
a1,a2...a₁, a₂ ...Powers (exponents) of each prime factor

Example substitution

360=23×32×5360 = 2^3 \times 3^2 \times 5

Divide 360 repeatedly: by 2 (three times), by 3 (twice), then by 5.

Worked examples

Find the prime factorisation of 60.

  1. 1
    60 ÷ 2: = 30
  2. 2
    30 ÷ 2: = 15
  3. 3
    15 ÷ 3: = 5
  4. 4
    5 is prime: stop
Answer: 2² × 3 × 5

Find the prime factorisation of 84.

  1. 1
    84 ÷ 2: = 42
  2. 2
    42 ÷ 2: = 21
  3. 3
    21 ÷ 3: = 7
  4. 4
    7 is prime: stop
Answer: 2² × 3 × 7

Find the prime factorisation of 100.

  1. 1
    100 ÷ 2: = 50
  2. 2
    50 ÷ 2: = 25
  3. 3
    25 ÷ 5: = 5
  4. 4
    5 is prime: stop
Answer: 2² × 5²

Frequently asked questions

What is prime factorisation?+

Prime factorisation expresses a number as a product of its prime factors. E.g., 60 = 2² × 3 × 5.

How do you find the prime factorisation of a number?+

Divide repeatedly by the smallest prime that goes in exactly, continuing with the result until you reach 1.

What is the prime factorisation of 360?+

360 = 2³ × 3² × 5.

Why is prime factorisation useful?+

It is used to find HCF and LCM, simplify fractions, and appears in exam questions on factors and multiples.

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