# Prime Factorisation

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When a number is expressed as a product of its factors we say that the number has been **factorized**. This is one of the factorisations of 24. The others are :

24 | ||||||||

2 | × | 12 | ||||||

2 | × | 6 | ||||||

2 | × | 3 | ||||||

24 | = | 2 | × | 2 | × | 2 | × | 3 |

24 | ||||||||

4 | × | 6 | ||||||

2 | × | 2 | 2 | × | 3 | |||

24 | = | 2 | × | 2 | × | 2 | × | 3 |

24 | ||||||||

3 | × | 8 | ||||||

2 | × | 4 | ||||||

2 | × | 2 | ||||||

24 | = | 3 | × | 2 | × | 2 | × | 2 |

In all the above factorisations of 24, we ultimately arrive at only one factorisation 2 × 2 × 2 × 3. In this factorisation the only factors 2 and 3 are *prime numbers*. Such a factorisation of a number is called a **prime factorisation**.

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Prime factorization is the process of finding which prime numbers multiply together to make the original number. A prime factorization tree is a useful tool for breaking down a number into its prime factors. Here are the basic rules and steps to create a prime factorization tree:

**Start with the Number:** Begin with the number you want to factorize.

**Find the Smallest Prime Factor:** Look for the smallest prime number that divides the given number. The prime numbers to check are 2, 3, 5, 7, 11, etc. Remember, 2 is the only even prime number.

**Divide the Number:** Divide your number by the smallest prime factor you found. The result of this division becomes the new number to factorize.

**Draw the Tree Branch:** In the prime factorization tree, draw a branch downward from your original number and write the prime factor you found. Underneath, write the result of your division.

**Repeat the Process:** Repeat steps 2 to 4 with the new number. Continue this process until the result is a prime number.

**End When You Reach a Prime Number and 1:** The process ends when the number you're dividing by is a prime number. This last prime number also becomes part of your prime factors alone with 1.

**The Prime Factorization:** The prime numbers at the ends of the branches in your tree are the prime factors of your original number. Multiply these prime factors together to check if they give you the original number.

**Handling Even Numbers:** If your number is even, you will often start with 2 as the first prime factor. Keep dividing by 2 until the result is odd.

**Handling Large Numbers:** For larger numbers, this process might take a while. Using divisibility rules can help find the prime factors more quickly.

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**Write the prime factorisations of 16, 28, 38 and 980.**

Prime factor of 16 =

Prime factor of 28 =

Prime factor of 38 =

Prime factor of 980 =