Factor trees: how to find prime factors step by step

Example: 60

1. Split 60 into two factors: $60 = 6 \times 10$.
2. Neither is prime, so split again: $6 = 2 \times 3$ and $10 = 2 \times 5$.
3. Circle the primes: 2, 3, 2, 5.
4. Multiply them in order: $60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$.

If you started with $60 = 4 \times 15$, you'd reach the same primes. Our prime factorisation calculator draws the tree for any number.

The primes under 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Remember 1 is not prime, and 2 is the only even prime.

Quick divisibility checks: even numbers divide by 2; if the digits add up to a multiple of 3, the number divides by 3; numbers ending in 0 or 5 divide by 5.

Write repeated primes as powers, smallest prime first: $2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$. Exam questions often say "as a product of its prime factors in index form".

• HCF and LCM: see how to find the HCF and the LCM.
• Square roots: halve the powers: $\sqrt{2^4 \times 3^2} = 2^2 \times 3 = 12$.
• Spotting squares and cubes: squares have all even powers; cubes have powers that are multiples of 3.

• Circling a number that isn't prime (9, 15, 21, 27).
• Calling 1 prime.
• Missing a branch when collecting the primes. Check by multiplying back to the original number.

Write as a product of primes in index form: (a) 72, (b) 150, (c) 252. Answers: $2^3 \times 3^2$, $2 \times 3 \times 5^2$, $2^2 \times 3^2 \times 7$. See the prime factorisation topic guide for more.

Start with any split: $360 = 36 \times 10$.

• $36 = 6 \times 6$, and each $6 = 2 \times 3$.
• $10 = 2 \times 5$.

The primes are 2, 3, 2, 3, 2 and 5. In index form: $360 = 2^3 \times 3^2 \times 5$. Check: $8 \times 9 \times 5 = 360$.

Some students prefer to divide by the smallest prime again and again:

So $84 = 2^2 \times 3 \times 7$. It gives the same answer as a tree, already in order.

To check whether a number is prime, try dividing by the primes up to its square root. For 97, $\sqrt{97} \approx 9.8$, so test 2, 3, 5 and 7. None divides exactly, so 97 is prime. Watch out for numbers that look prime but aren't: $91 = 7 \times 13$, $51 = 3 \times 17$ and $57 = 3 \times 19$.