Break every factor until only primes remain

Every whole number greater than 1 is either prime or composite. A composite number has a factor other than 1 and itself, so it splits into a factor pair, and each composite factor splits again.

The tree ends when every branch ends at a prime factor, and the product of the branch ends is the number at the start:

Different trees finish with the same primes

84 can begin as 12 × 7, 6 × 14 or 4 × 21:

Repeated division collects the same factors

Divide 756 by the smallest prime that divides exactly, then repeat with the quotient:

1. Divide by the smallest prime that divides exactly.
2. Repeat with the quotient, testing the same prime again before moving to the next.
3. Stop when the quotient is 1; the divisors down the side are all the prime factors.

Collect repeated primes in index form

An index of 1 is not written: the final 7 occurs once, so the answer ends in × 7, not × 7^¹. The completed answer is in index form, with its prime bases in ascending order so that two factorisations can be compared prime by prime. “Write 756 as a product of its prime factors” asks for this form.

Factorise a chosen number

A factor tree can be built from any composite branch, and a branch can be split again or collapsed. The final leaves show the prime factorisation of the starting number.

Common mistakes

• Stopping at a composite factor Writing 90 = 9 × 10 leaves 9 and 10 unsplit, so it is a factor pair rather than a prime factorisation.
• Losing a repeated factor Writing 72 = 2^² × 3^² gives 36; the complete factorisation is 2^³ × 3^².
• Using a composite base Writing 72 = 8 × 3^² multiplies the three 2s out to 8, which is composite; the index form keeps them as 2^³.

Summary

1. Split a composite number into factor pairs, or divide it repeatedly by primes.
2. Continue until every remaining factor is prime, or the repeated-division quotient is 1.
3. Every whole number greater than 1 has one prime factorisation, whichever tree or division reaches it.
4. Keep every repeated copy of a prime factor.
5. Write equal prime factors as powers, with prime bases in ascending order.
6. Multiply the prime powers to recover the starting number and check the factorisation.