Prime Factorization of Integers

What is prime factorization?

Every integer greater than 1 can be written as a product of prime numbers in exactly one way (up to order). This is the Fundamental Theorem of Arithmetic. Finding this product is called prime factorization.

Step-by-step: Factor 360

Method 1 — Repeated division by the smallest prime:

Step 1: Is 360 divisible by 2? Yes: $360 \div 2 = 180$.

Step 2: Is 180 divisible by 2? Yes: $180 \div 2 = 90$.

Step 3: Is 90 divisible by 2? Yes: $90 \div 2 = 45$.

Step 4: Is 45 divisible by 2? No. Try 3: $45 \div 3 = 15$.

Step 5: Is 15 divisible by 3? Yes: $15 \div 3 = 5$.

Step 6: 5 is prime. Stop.

Result: $360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^3 \times 3^2 \times 5$

$\boxed{360 = 2^3 \times 3^2 \times 5}$

Verification: $8 \times 9 \times 5 = 72 \times 5 = 360$ ✓

Method 2 — Factor tree

Start with 360 and split into any two factors. Keep splitting until all leaves are prime:

$360 = 36 \times 10 = (4 \times 9) \times (2 \times 5) = (2 \times 2) \times (3 \times 3) \times 2 \times 5$

No matter how you split, you always get the same primes: $2^3 \times 3^2 \times 5$.

How to know when you're done

Stop testing primes when you reach a prime $p$ where $p^2 >$ the remaining number. For example, if the remaining number is 37: $7^2 = 49 > 37$, so 37 must be prime (no prime up to 6 divides it).

The primes to test

$2, 3, 5, 7, 11, 13, 17, 19, 23, \ldots$ You only need to test primes up to $\sqrt{n}$.

Number of divisors

From $360 = 2^3 \times 3^2 \times 5^1$: the number of divisors is $(3+1)(2+1)(1+1) = 4 \times 3 \times 2 = 24$. This counts all combinations of prime powers from $2^0$ through $2^3$, $3^0$ through $3^2$, and $5^0$ through $5^1$.

Common mistakes

• Forgetting to keep dividing by the same prime. $360 \div 2 = 180$, but don't move to 3 yet — try 2 again: $180 \div 2 = 90$, $90 \div 2 = 45$. Only move to 3 when 2 no longer divides.
• Thinking 1 is prime. It's not. The factorization doesn't include 1.

Try it in the visualization

Enter any number. The factor tree builds step by step — each split shows two children, and leaves (primes) are highlighted. The final factorization is displayed in exponential form.