Solution: The LCM of 105 and 45 is 315
Methods
How to find the LCM of 105 and 45 using Prime Factorization
One way to find the LCM of 105 and 45 is to start by comparing the prime factorization of each number. To find the prime factorization, you can follow the instructions for each number here:
What are the Factors of 105?
What are the Factors of 45?
Here is the prime factorization of 105:
3
1
×
5
1
×
7
1
3^1 × 5^1 × 7^1
3 1 × 5 1 × 7 1
And this is the prime factorization of 45:
3
2
×
5
1
3^2 × 5^1
3 2 × 5 1
When you compare the prime factorization of these two numbers, you want to look for the highest power that each prime factor is raised to. In this case, there are these prime factors to consider: 3, 5, 7
3
2
×
5
1
×
7
1
=
315
3^2 × 5^1 × 7^1 = 315
3 2 × 5 1 × 7 1 = 315
Through this we see that the LCM of 105 and 45 is 315.
How to Find the LCM of 105 and 45 by Listing Common Multiples
The first step to this method of finding the Least Common Multiple of 105 and 45 is to begin to list a few multiples for each number. If you need a refresher on how to find the multiples of these numbers, you can see the walkthroughs in the links below for each number.
Let’s take a look at the multiples for each of these numbers, 105 and 45:
What are the Multiples of 105?
What are the Multiples of 45?
Let’s take a look at the first 10 multiples for each of these numbers, 105 and 45:
First 10 Multiples of 105: 105, 210, 315, 420, 525, 630, 735, 840, 945, 1050
First 10 Multiples of 45: 45, 90, 135, 180, 225, 270, 315, 360, 405, 450
You can continue to list out the multiples of these numbers as long as needed to find a match. Once you do find a match, or several matches, the smallest of these matches would be the Least Common Multiple. For instance, the first matching multiple(s) of 105 and 45 are 315, 630, 945. Because 315 is the smallest, it is the least common multiple.
The LCM of 105 and 45 is 315.
Find the LCM of Other Number Pairs
Want more practice? Try some of these other LCM problems:
What is the LCM of 29 and 117?
What is the LCM of 14 and 103?
What is the LCM of 97 and 147?
What is the LCM of 19 and 49?
What is the LCM of 72 and 123?