Prime Factor Decomposition

Prime Factor Decomposition might sound like a mouthful, but it’s really just a way of breaking down numbers into their smallest "building blocks." These building blocks are prime numbers. Think of it like taking apart a Lego tower and seeing the individual pieces that made it up!

 

What is a Prime Number?

A prime number is a number that only has two factors: 1 and itself. It can’t be divided evenly by any other number. For example:

• $2, 3, 5, 7  and 11$ are all prime numbers
• Numbers like $4  or  6$ are not prime because they have more than two factors.

 

What is Prime Factor Decomposition?

Prime Factor Decomposition (sometimes called "prime factorization") is breaking down a number into a product of prime numbers. In other words, we find all the prime numbers that multiply together to make the original number.

Here’s a simple way to decompose a number into prime factors using a method called the "factor tree." We’ll do this step by step:

1. Start with the Number: Write down the number you’re working with at the top.
2. Find a Pair of Factors: Split the number into any two factors that multiply to give the original number.
3. Repeat with Each Branch: Keep breaking down each number until you’re left with prime numbers.
4. Circle the Primes: Once you have only prime numbers, you’re done!

 

Example

Decompose 60 into Prime Factors

Let's break down $60$ into its prime factors using a factor tree:

1. Start with 60:
   Choose two factors of $60$ that multiply to make $60$. For example: $$60 = 6 \times 10$$

2. Decompose 6 and 10: 
   • $6$ can be split inyo $2 \times 3$
   • $10$ can be split into $2 \times 5$
   • So, the factor tree looks like this:

 

 

1. Circle the Prime Numbers:
   We now have all prime numbers: $2,3,2  and 5$.
2. Write the Answer in Multiplicative Form:
   • Combine the circled prime numbers as a product: $$60 = 2 \times 2 \times 3 \times 5$$
   • Or, we can write it using exponents (to show how many times each prime is used): $$60 = 2^2 \times 3 \times 5$$