Deep Dive: Prime Factorization

The Goal: Every whole number greater than 1 is either a prime number itself, or it can be broken down into a specific list of prime numbers multiplied together.

This list is unique—like a numerical fingerprint. Finding this list is called Prime Factorization.

Remember your first few primes: 2, 3, 5, 7, 11, 13, 17...

The Golden Rule: Index Form

When you find the prime factors, you must write the final answer using powers (indices). Don't just leave a long list of numbers.

If you find: \(2 \times 2 \times 2 \times 3 \times 5 \times 5\)
Write it as: \(2^3 \times 3 \times 5^2\)

Method 1: The Factor Tree (Visual)

This is the most popular method. You keep splitting numbers into pairs of factors until you only have primes left.

Example A: Find the prime factors of 120

Strategy: Split 120 into any two numbers. Circle primes as you find them. Keep splitting the others.

Start: 120. Let's split it into \(12 \times 10\).
Split 12 into \(4 \times 3\). 3 is prime! Circle it.
Split 4 into \(2 \times 2\). 2 and 2 are prime! Circle them.
Now back to the 10. Split 10 into \(2 \times 5\). 2 and 5 are prime! Circle them.
Every branch ends in a circled prime. We are done.

Collect all the circled primes: \(2, 2, 2, 3, 5\).

\(120 = 2^3 \times 3 \times 5\)

Method 2: Repeated Division (The Ladder)

This method is tidier for larger numbers. You keep dividing by the smallest prime number possible until you reach 1.

Example B: Find the prime factors of 252

Strategy: Draw a line. Put the number on the right. Divide by small primes (start with 2) and put the result underneath.

We reached 1, so we stop. Collect the primes used: \(2, 2, 3, 3, 7\).

\(252 = 2^2 \times 3^2 \times 7\)

Does the order matter?

No! With a factor tree, you could start 120 as \(2 \times 60\), or \(10 \times 12\), or \(4 \times 30\). As long as your math is correct, you will always end up with the exact same collection of prime numbers at the end.

Your Turn: Practice Questions

Find the prime factorization of the following numbers. Give your answers in index form.

Click on the question bar to reveal the worked solution.

Question 1: Write 64 as a product of prime factors.

Method: Repeated Division by 2

Since 64 is even, we just keep dividing by 2.

\(64 \div \mathbf{2} = 32\)
\(32 \div \mathbf{2} = 16\)
\(16 \div \mathbf{2} = 8\)
\(8 \div \mathbf{2} = 4\)
\(4 \div \mathbf{2} = 2\)
\(2 \div \mathbf{2} = 1\) (Stop)

We have six 2s.

\(64 = 2^6\)
Question 2: Find the prime factorization of 150.

Method: Factor Tree approach

1. Split 150 into \(15 \times 10\).

2. Split 15 into \(\mathbf{3} \times \mathbf{5}\). (Both are prime, circle them).

3. Split 10 into \(\mathbf{2} \times \mathbf{5}\). (Both are prime, circle them).

Collecting the primes: \(2, 3, 5, 5\).

\(150 = 2 \times 3 \times 5^2\)
Question 3: Write 98 as a product of primes in index form.

Method: Repeated Division

1. It's even, divide by 2: \(98 \div \mathbf{2} = 49\).

2. 49 isn't divisible by 2, 3, or 5. Try 7. \(49 \div \mathbf{7} = 7\).

3. 7 is prime. \(7 \div \mathbf{7} = 1\).

Collecting primes: \(2, 7, 7\).

\(98 = 2 \times 7^2\)
Question 4: Find the prime factorization of 360.

Method: Factor Tree approach

This is a bigger number, try splitting off the zero first: \(360 = 36 \times 10\).

Branch 1 (The 10): Split into \(\mathbf{2} \times \mathbf{5}\). (Done).

Branch 2 (The 36): Split into \(6 \times 6\).

Split the first 6 into \(\mathbf{2} \times \mathbf{3}\). (Done).

Split the second 6 into \(\mathbf{2} \times \mathbf{3}\). (Done).

Collect all primes found: \(2, 5, 2, 3, 2, 3\).

Group them: Three 2s, two 3s, one 5.

\(360 = 2^3 \times 3^2 \times 5\)
Question 5: Write 625 as a product of prime factors.

Method: Recognition / Repeated Division

It ends in 5, so we know 5 is a prime factor.

\(625 \div \mathbf{5} = 125\)
\(125 \div \mathbf{5} = 25\)
\(25 \div \mathbf{5} = 5\)
\(5 \div \mathbf{5} = 1\) (Stop)

We found four 5s.

\(625 = 5^4\)