A prime number has exactly two factors, 1 and itself. Every other whole number greater than 1 is composite: it splits into smaller factors. Keep splitting, and you reach the primes that build the number.
Key idea
To find the prime factorization, split the number into two factors, then split any factor that is not prime. Stop only when every branch ends in a prime. Then write the number as the product of those primes.
Why it works
Each split makes the factors smaller, so the splitting has to stop, and it can only stop at primes. However you begin, you end with the same primes: $60 = 6 \times 10$ and $60 = 4 \times 15$ both lead to $2 \times 2 \times 3 \times 5$.
Worked examples
Worked example: A factor tree for 60
Write 60 as a product of primes.
- $60 = 6 \times 10$. Neither 6 nor 10 is prime.
- $6 = 2 \times 3$ and $10 = 2 \times 5$. All four are prime.
- Put them in order: $2 \times 2 \times 3 \times 5$. Check: $4 \times 3 \times 5 = 60$.
Answer: $60 = 2 \times 2 \times 3 \times 5$
Worked example: Peel off small primes
Write 126 as a product of primes.
- 126 is even: $126 = 2 \times 63$.
- 63 is not even, but $6 + 3 = 9$, so 3 divides it: $63 = 3 \times 21$.
- $21 = 3 \times 7$. Now every factor is prime.
Answer: $126 = 2 \times 3 \times 3 \times 7$
Common mistake
Stopping at a composite factor, like writing $60 = 2 \times 2 \times 15$. The number 15 is not prime ($15 = 3 \times 5$), so the tree is not finished. Check each factor: can it be divided by 2, 3, 5 or 7 without leaving a remainder? If so, split it.
You can write repeated primes with an exponent, like $2^{2} \times 3 \times 5$. You will meet exponents properly with powers of ten.