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1.4 E) Expressing Numbers as a Product of their Prime Factors

1.4 E) Expressing Numbers as a Product of their Prime Factors

A prime number is a number which is only divisible by itself and 1 (click here for more information about prime numbers). We are able to express a number as a product of its prime factors, which are the prime numbers that when multiplied together give us the number that we are expressing as a product of its prime factors.

The easiest way to express a number as a product of its prime factors is to use a prime factor tree. The top of the factor tree is the number that we are expressing as a product of its prime factors. We then find a pair of numbers that multiply together to give this number and we have two branches with these numbers at the end. If any of these two number are prime, we circle them and leave them. If any of the two numbers are not prime, we find two factors that go into this number. We continue this process until all of the branches end with prime numbers (circled number).

It does not matter which two factors we use first because we will still obtain the same prime factors.

Let’s have a few examples.

The easiest way to express a number as a product of its prime factors is to use a prime factor tree. The top of the factor tree is the number that we are expressing as a product of its prime factors. We then find a pair of numbers that multiply together to give this number and we have two branches with these numbers at the end. If any of these two number are prime, we circle them and leave them. If any of the two numbers are not prime, we find two factors that go into this number. We continue this process until all of the branches end with prime numbers (circled number).

It does not matter which two factors we use first because we will still obtain the same prime factors.

Let’s have a few examples.

Example 1

Express 12 as a product of its prime factors.

We start this question with a 12 at the top. We then choose two numbers that multiply together to give 12. I am going to choose 4 and 3, so I create two branches with 4 and 3 at the end of each of them.

Express 12 as a product of its prime factors.

We start this question with a 12 at the top. We then choose two numbers that multiply together to give 12. I am going to choose 4 and 3, so I create two branches with 4 and 3 at the end of each of them.

I now check both of these numbers to see whether any of them are prime and circle any prime factors. 3 is prime because it can only be divided by itself (3) and 1. Therefore, I circle 3. 4 is not prime.

I now find two factors that go into 4, which are 2 and 2.

2 is prime so I circle both of these factors. All of the branches above have ended in circled numbers and this means that I have found all the prime factors.

The final step is to write 12 as a product of its prime factors, which is done by writing all of the prime factors above multiplied by together.

The final step is to write 12 as a product of its prime factors, which is done by writing all of the prime factors above multiplied by together.

Earlier I said that it does not matter what factors I choose first. Originally, I chose the factors 4 and 3. I am now going to complete the factor tree again but with a different initial pair; the initial pair that I am going to use is 6 and 2.

2 is prime so I circle it. 6 is not prime, so I look for two factors that I multiply together to give 6. The two factors are 2 and 3.

Both 2 and 3 are prime, so I circle them. All of the branches have circled numbers at the end of them and this means that I have found all of the prime factors. I now need to express 12 as a product of its prime factors, which is the same as above:

Therefore, you can see that it does not matter which pair of factors we start off with as you will always get the same number of each prime numbers.

Example 2

Express 84 as a product of its prime factors.

The tree starts with 84 at the top. If the number that we are expressing as a product of its prime factors is even, it is usually a good idea to just divide by 2 and keep dividing by 2 until you can no longer divide by 2. When you can no longer divide by 2, you can look for other factors.

84 divided by 2 is 42. Therefore, the first pair of factors that we are going to have is 2 and 42.

Express 84 as a product of its prime factors.

The tree starts with 84 at the top. If the number that we are expressing as a product of its prime factors is even, it is usually a good idea to just divide by 2 and keep dividing by 2 until you can no longer divide by 2. When you can no longer divide by 2, you can look for other factors.

84 divided by 2 is 42. Therefore, the first pair of factors that we are going to have is 2 and 42.

2 is prime, so we circle it. 42 is not prime, which means that we need to find two factors that multiply together to give us 42. 42 is even, which means that we can divide 42 by 2 to find the factor pair (42 ÷ 2 = 21). Therefore, a factor pair for 42 is 2 and 21.

2 is prime, so we circle it. 21 is not prime. It is also not even, which means that 2 will not be a factor. A factor pair of 21 is 3 and 7.

Both 3 and 7 are prime, which means that we circle both of them. All of the branches have circled numbers at the end of them, which means that we have found all of the prime factors for 84. Therefore, 84 as a product of its prime factors is:

We can write this in index form as: