How to find the LCM with prime factorization?
Understand the Problem
The question is asking for a method to find the least common multiple (LCM) of numbers using prime factorization, which involves breaking down each number into its prime factors and then using those to determine the LCM.
Answer
$36$
Answer for screen readers
The least common multiple (LCM) of 12 and 18 is $36$.
Steps to Solve
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Identify the numbers First, you need to identify the numbers for which you want to find the LCM. Let's say we want to find the LCM of 12 and 18.
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Prime factorization Next, you perform prime factorization on each number.
- For 12:
- $12 = 2^2 \times 3^1$ (the prime factors of 12 are 2 and 3)
- For 18:
- $18 = 2^1 \times 3^2$ (the prime factors of 18 are also 2 and 3)
- Determine the highest powers of each prime factor Now, you need to take the highest power of each prime factor found in the previous step.
- The prime factor 2 has the highest power of $2^2$ from 12.
- The prime factor 3 has the highest power of $3^2$ from 18.
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Calculate the LCM Multiply these highest powers together to find the LCM: $$ LCM(12, 18) = 2^2 \times 3^2 $$
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Final calculation Now compute the product: $$ 2^2 = 4 \quad \text{and} \quad 3^2 = 9 $$ So, $$ LCM(12, 18) = 4 \times 9 = 36 $$
The least common multiple (LCM) of 12 and 18 is $36$.
More Information
The LCM is the smallest number that is a multiple of both given numbers. In this case, $36$ is the first multiple that both $12$ and $18$ share. Prime factorization is an effective method for finding the LCM, especially for larger numbers.
Tips
- Forgetting to compare powers: Make sure to compare the powers of prime factors correctly and select the highest.
- Not including all prime factors: Remember that each prime factor must be represented, even if one of the numbers does not contain it.
