How to find LCM by prime factorisation?
Understand the Problem
The question is asking for a method to find the least common multiple (LCM) using prime factorization. This involves breaking down the numbers into their prime factors and then using those factors to determine the LCM.
Answer
The LCM of 12 and 15 is $60$.
Answer for screen readers
The least common multiple of 12 and 15 is $60$.
Steps to Solve
- Identify the numbers to find the LCM for
First, list the numbers for which you want to find the least common multiple. For example, let's say we want to find the LCM of 12 and 15.
- Prime factorization of each number
Next, perform prime factorization on each of the numbers.
- For 12: $$ 12 = 2^2 \times 3^1 $$
- For 15: $$ 15 = 3^1 \times 5^1 $$
- Identify the highest powers of each prime factor
Now, identify all the prime factors involved and take the highest power for each:
- From 12:
- $2^2$
- $3^1$
- From 15:
- $3^1$
- $5^1$
The highest powers are:
- $2^2$ from 12
- $3^1$ from both
- $5^1$ from 15
- Multiply the highest powers together
Finally, multiply the highest powers of each prime factor together to get the LCM: $$ \text{LCM} = 2^2 \times 3^1 \times 5^1 $$
Calculating this gives: $$ \text{LCM} = 4 \times 3 \times 5 = 60 $$
The least common multiple of 12 and 15 is $60$.
More Information
The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both. Using prime factorization is an efficient way to ensure you've accounted for all factors correctly, especially with larger numbers.
Tips
- Not including all prime factors: Make sure to include all unique prime factors from both numbers.
- Using incorrect powers: Always take the highest power of each prime factor from the factorizations to ensure you are finding the LCM properly.
