How do you find the lowest common multiple between two numbers quickly and how can you check your answer?
The lowest common multiple (LCM) between the two numbers 8 and 10 is 40. The lowest common multiple is the smallest number that both 8 and 10 can be divided into without leaving a remainder.
Is there an easy way to find the lowest common multiple between two numbers?
There are several ways you can find the LCM between two numbers. One way is to just guess – this isn’t the most effective way, as there are a lot of numbers you could guess! Another is to start from the number 2 and then divide both the smaller number and the larger number into this number and if it doesn’t work to continue incrementing the number until you do find the answer – this will work, but isn’t the most efficient way. The third way is to use multiples of the larger number and then divide the smaller number into each multiple. This is the most effective and efficient way to get to find the lowest common multiple .
Here is a step by step guide demonstrating how you can get the lowest common multiple between two numbers using the third approach – the most efficient means.
Step By Step To Find LCM Quickly
To find the lowest common multiple between 2 numbers find the smaller number of the two numbers and ask yourself the question: can the smaller number be divided into the larger number without leaving a remainder? (Remember this question as you will be asking it to yourself frequently!)
If the smaller number can be divided into the larger number then you have your lowest common multiple of the two numbers and you can stop (yay!). If the smaller number cannot be divided into the larger number then you need to increase the larger number, and this is done through multiples of the larger number. Therefore, the next step will be to double the larger number (multiply it by 2) and with your new larger number to ask yourself that same question again: can the smaller number be divided into the new larger number without leaving a remainder?
When you answer Yes! you can stop as you have found your lowest common multiple. However, every time you answer No! you need to increase the larger number again and repeat the questioning process. Therefore, if you were still going you would now be tripling the size of your large number (multiplying it by 3) and with your new larger number asking yourself that same question again.
This process continues until you finally have found a multiple of the larger number that the smaller number can be divided into without leaving a remainder.
If you never get to answer Yes! then you can stop when the number used to increase the large number (the multiplier) ends up being the same number as the smaller number .
To help demonstrate this process I’ll use an example.
Finding LCM Of 8 And 10
To find the lowest common multiple between two numbers you need to know if both numbers can be divided into a larger number without leaving a remainder. The process is simple and repetitive. The hardest part is making sure you’ve correctly performed the division of the smaller number into the multiples of the larger numbers.
What is the lowest common multiple for the numbers 8 and 10? Using the process above this would look as follows:
- Does 8 divide into 10 without leaving a remainder?
- No, there is a remainder of 2.
- Increment the multiplier by 1 and multiply it to the larger number 10 × 2 = 20
- Does 8 divide into 20 without leaving a remainder?
- No, there is a remainder of 4.
- Increment the multiplier by 1 and multiply it to the larger number 10 × 3 = 30
- Does 8 divide into 30 without leaving a remainder?
- No, there is a remainder of 6.
- Increment the multiplier by 1 and multiply it to the larger number 10 × 4 = 40
- Does 8 divide into 40 without leaving a remainder?
- Yes, 8 goes 5 times into 40 with no remainder.
Therefore, the lowest common multiple between 8 and 10 is 40.
How To Check Your Answer
Another way of finding the lowest common multiple, and a way to double check your answer, is to start by multipying both numbers together and working backwards. This can take longer if the smaller number is a large number itself.
Trying this with the same two numbers above multiplying the two numbers together 8 and 10 produces the common multiple of 80. However, note this multiple might not be the lowest common multiple and the only way to confirm is to count backwards.
With the previous method the multiplier started at 1 and incremented up, but with this approach the multiplier starts from the smaller number and decrements down to 1. Here’s how this process works:
- Find the smaller of the two numbers, this would be 8.
- Decrement the smaller number down by 1. This would mean the multiplier is 7.
- Multiply this multiplier to the bigger number 10 × 7 = 70 .
- Does the smaller number 8 divide into 70 without leaving a remainder?
- No. Dividing 8 into 70 leaves a remainder of 6.
- Decrement the smaller number down by 1. This would mean the multiplier is 6.
- Multiply this multiplier to the bigger number 10 × 6 = 60 .
- Does the smaller number 8 divide into 60 without leaving a remainder?
- No. Dividing 8 into 60 leaves a remainder of 4.
- Decrement the smaller number down by 1. This would mean the multiplier is 5.
- Multiply this multiplier to the bigger number 10 × 5 = 50 .
- Does the smaller number 8 divide into 50 without leaving a remainder?
- No. Dividing 8 into 50 leaves a remainder of 2.
- Decrement the smaller number down by 1. This would mean the multiplier is 4.
- Multiply this multiplier to the bigger number 10 × 4 = 40 .
- Does the smaller number 8 divide into 40 without leaving a remainder?
- Yes, 8 goes into 40 without leaving a remainder. Therefore, write this number down and continue to see if there is another multiple less than this.
- Decrement the smaller number down by 1. This would mean the multiplier is 3.
- Multiply this multiplier to the bigger number 10 × 3 = 30 .
- Does the smaller number 8 divide into 30 without leaving a remainder?
- No. Dividing 8 into 30 leaves a remainder of 6.
- Decrement the smaller number down by 1. This would mean the multiplier is 2.
- Multiply this multiplier to the bigger number 10 × 2 = 20 .
- Does the smaller number 8 divide into 20 without leaving a remainder?
- No. Dividing 8 into 20 leaves a remainder of 4.
- Decrement the smaller number down by 1. This would mean the multiplier is 1.
- Multiply this multiplier to the bigger number 10 × 1 = 10 .
- Does the smaller number 8 divide into 10 without leaving a remainder?
- No. Dividing 8 into 10 leaves a remainder of 2.
Now that you’ve finished you will end up with a list of numbers like so: (80, 40)
With this list it’s just a simple matter of choosing the smallest, which should be the last number: 40
As you can see once you have all of the common multiples between 8 and 10 you just need to find the smallest number as this represents the lowest common multiple (LCM), and in this example this ends up being 40.
Using both approaches should help you quickly find and confirm the lowest common multiple between your two numbers.
Summary
To find the lowest common multiple (LCM) between two numbers, namely 8 and 10, ask yourself if the smaller number can be divided into the larger number without leaving a remainder. If it can you have your LCM, if not increase the larger number by an incremental multiplier (2, 3, 4, 5… etc) and repeat.
Knowing how to find the lowest common multiple will help you with arithemtic math problems involving the addition and/or subtraction of fractions.