How to find LCM with prime factorization?
Understand the Problem
The question is asking for a method to find the least common multiple (LCM) of numbers using the prime factorization method. This involves breaking down the numbers into their prime factors and applying specific rules to determine the LCM.
Answer
The LCM of 12 and 18 is 36.
Answer for screen readers
The least common multiple (LCM) of 12 and 18 is 36.
Steps to Solve
- Factor the Numbers into Primes
Start by breaking down each number into its prime factors. For example, if you have the numbers 12 and 18:
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Factor 12: $$ 12 = 2^2 \times 3^1 $$
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Factor 18: $$ 18 = 2^1 \times 3^2 $$
- Identify Unique Prime Factors
List all unique prime factors from the factorizations you've obtained. For the above example, the unique prime factors are 2 and 3.
- Determine the Highest Powers of Each Prime
For each unique prime factor, take the highest power that appears in the factorizations:
- For the prime factor 2, the highest power is $2^2$ (from 12).
- For the prime factor 3, the highest power is $3^2$ (from 18).
- Multiply the Highest Powers to Find the LCM
Now multiply all the highest powers together to get the LCM:
$$ LCM = 2^2 \times 3^2 $$
Calculating this gives:
$$ LCM = 4 \times 9 = 36 $$
The least common multiple (LCM) of 12 and 18 is 36.
More Information
The least common multiple is useful in various applications, such as finding a common denominator for fractions or scheduling events that repeat at different intervals. It ensures that you have a mutual timing for cycles represented by the numbers.
Tips
- Forgetting to include all unique prime factors, which can lead to an incorrect LCM.
- Not using the highest power of prime factors, which alters the correct calculation of the LCM.
