Prior to observing how to find the lowest common multiple, LCM by prime factorization, we can look at a reminder of what prime factors are and how to find them.
Prime Factorization of Numbers
Let’s look at a number for an example, we can try 36.
An effective method for prime factorization of 36, is to keep carrying out division by prime numbers, until the result of the division ends up being a prime number.
Start off by dividing by the lowest prime number there is, which is 2.
2) 18 ÷ 2 = 9
3) 9 ÷ 2 = 4.5 Doesn’t divide exactly, so start with the next prime number after 2 instead, 3.
4) 9 ÷ 3 = 3
3 is a prime number, so we can stop our dividing here.
Now, the different prime numbers that were used to divide at each stage when multiplied together along with the final prime number obtained from the division, should give back the original number that we began with, 36.
In this case with 36, 2 was used a total of 2 times for exact division, and 3 one time before we eventually obtained the prime number of 3.
This form can also be simplified further with an exponent/power, to give the following. 22 × 32 = 36
In either form, ti can be seen that 2 and 3 are factors that are prime.
That concludes the prime factorization for 36.
Find LCM by Prime Factorization
We can now look at how we can find the lowest common multiple, LCM by prime factorization.
The Approach:
The find GCF by Prime Factorization page showed an example of who to find the greatest common factor using prime factors.This is initially how we can approach finding the lowest common multiple, LCM also.
Example (1.1) on that page involved finding the GCF of the numbers 18 and 78.
We’ll show that example again here and how it leads us to find the LCM of 18 and 78.
First prime factorizing both numbers separately.
1) 18 ÷ 2 = 9
2) 9 ÷ 2 = 4.5 Doesn’t divide exactly.
3) 9 ÷ 3 = 3 PRIME NUMBER
1) 78 ÷ 2 = 39
2) 39 ÷ 2 = 19.5 Doesn’t divide exactly.
3) 39 ÷ 3 = 13 PRIME NUMBER
78 ) 2 × 3 × 13 = 78
The prime factors that are shared, when multiplied together, will give the greatest common factor (GCF), of the two numbers.
When prime factorized, the numbers 18 and 78 both share one 2, and one 3.
2 × 3 = 6 => The greatest common factor of 18 and 78 is 6.
Now, this GCF multiplied by the prime factors that are NOT shared, will give the lowest common multiple, LCM.
From 18. One 3 was NOT shared.
From 18. Only 13 was NOT shared.
So we can do the following multiplication. 3 × 13 × 6 = 234
The lowest common multiple of 18 and 78 is 234.
Find LCM by Prime Factorization Summary
1) Prime factorize both numbers.
2) Work out the greatest common factor, GCF.
3) GCF multiplied by all prime factors NOT used to find GCF, will find lowest common multiple, LCM.
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