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1.4 G) Highest Common Factor & Lowest Common Multiple – Part 2
1.4 G) Highest Common Factor & Lowest Common Multiple – Part 2
The lowest common multiple is the lowest number that is a multiple of two or more numbers; it is the first/ lowest number that two or more number go into.
There are two different methods that we can use to work out the lowest common multiple. These are:
The examples in this question build on the examples that we have looked at in the previous section. Therefore, make sure that you have gone through the previous section before you work through this section (click here to be taken through to the previous section).
There are two different methods that we can use to work out the lowest common multiple. These are:
- Write down all of the multiples for the numbers involved and find the lowest multiple that is in both of the lists. This method is easy when we are finding the lowest common multiple for small numbers, but it is not the best method to use when we are finding the lowest common multiple for larger numbers.
- Use factor trees.
The examples in this question build on the examples that we have looked at in the previous section. Therefore, make sure that you have gone through the previous section before you work through this section (click here to be taken through to the previous section).
Example 1
What is the lowest common multiple for 6 and 8?
Method 1 – Writing all of the Multiples
For the first method, we are just going to write the multiples out (the multiples of 6 and the multiples of 8). We will then find the lowest number that is a multiple of both numbers. You write the multiples out for the first number up to an appropriate point. There is no need to write loads of multiples out because you will just waste time. Also, you can always write more multiplies out if there is not a multiple that is in both lists.
What is the lowest common multiple for 6 and 8?
Method 1 – Writing all of the Multiples
For the first method, we are just going to write the multiples out (the multiples of 6 and the multiples of 8). We will then find the lowest number that is a multiple of both numbers. You write the multiples out for the first number up to an appropriate point. There is no need to write loads of multiples out because you will just waste time. Also, you can always write more multiplies out if there is not a multiple that is in both lists.
24 is the lowest number that appears in both of the lists, which means that the lowest common multiple between 6 and 8 is 24.
Method 2 – Factor Trees
The factor tree method involves expressing the two numbers as a product of their prime factors. We then find the highest common factor, which is the largest number that goes into both of the numbers.
We then need to cross out any factors that appear in both of the numbers (the factors that we have used to find the highest common factor). The final step is to multiply the highest common factor by the rest of the factors for the two numbers. This will all make much more sense when we work through the example.
The first step is to use factor trees to express the two numbers as a product of their prime factors.
Method 2 – Factor Trees
The factor tree method involves expressing the two numbers as a product of their prime factors. We then find the highest common factor, which is the largest number that goes into both of the numbers.
We then need to cross out any factors that appear in both of the numbers (the factors that we have used to find the highest common factor). The final step is to multiply the highest common factor by the rest of the factors for the two numbers. This will all make much more sense when we work through the example.
The first step is to use factor trees to express the two numbers as a product of their prime factors.
The highest common factor is found by multiplying any common prime factors (if there are more than 2 common prime factors). There is only one common factor between 6 and 8, which is 2. Therefore, the highest common factor between 6 and 8 is 2.
Before we are able to obtain the lowest common multiple, we cross out any factors that are common (the common ones have been used to find the highest common factor). We then multiply the highest common factor by all of the factors that are left for the two numbers.
The only common factor was 2, which means that I will cross out one 2 from each of the prime factor lists.
The only common factor was 2, which means that I will cross out one 2 from each of the prime factor lists.
We obtain the lowest common multiple by multiplying the highest common factor by all of the factors that are left.
Therefore, the lowest common multiple for 6 and 8 is 24.
For smaller numbers, it is probably easier to use the first method because the second method is considerably more time consuming. However, when the numbers are larger, it is probably easier to use the second method because it will take a while to write down the multiples of large numbers.
For smaller numbers, it is probably easier to use the first method because the second method is considerably more time consuming. However, when the numbers are larger, it is probably easier to use the second method because it will take a while to write down the multiples of large numbers.
Example 2
What is the lowest common multiple for 42 and 63?
I am only going to use the factor tree method to work out the answer for this question. Below are the factor trees and the two numbers expressed as a product of their prime factors.
What is the lowest common multiple for 42 and 63?
I am only going to use the factor tree method to work out the answer for this question. Below are the factor trees and the two numbers expressed as a product of their prime factors.
The highest common factor is obtained by multiplying common factors. There are two common prime factors between 42 and 63 and these are 3 and 7. Therefore, the highest common factor for 42 and 63 is 21 (3 x 7).
Before we are able to obtain the lowest common multiple we need to cross out the common factors.
We then multiply the highest common factor by the rest of the factors that are left.
Therefore, the lowest common multiple for 42 and 63 is 126.
We could have found the lowest common multiple by writing down all of the multiples for 42 and 63. We would then find the lowest number that appears in both of the lists. However, it would be very easy to make mistakes because I doubt that many of us know our 42 and 63 multiple tables off by heart!
We could have found the lowest common multiple by writing down all of the multiples for 42 and 63. We would then find the lowest number that appears in both of the lists. However, it would be very easy to make mistakes because I doubt that many of us know our 42 and 63 multiple tables off by heart!