LCM (Least Common Multiple) prime factorization is a method to find the smallest number that is a multiple of two or more numbers by breaking down each number into its prime factors. Here's how it works:
Steps for Finding the LCM Using Prime Factorization:
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Prime Factorization: Find the prime factorization of each number. This means expressing each number as a product of its prime factors. A prime number is a number greater than 1 that has only two factors: 1 and itself (e.g., 2, 3, 5, 7, 11).
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Identify All Prime Factors: List all the prime factors that appear in any of the prime factorizations.
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Highest Power: For each prime factor, find the highest power (exponent) to which it appears in any of the prime factorizations.
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Multiply: Multiply together each prime factor raised to its highest power found in step 3. The result is the LCM.
Example: Finding the LCM of 12 and 18
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Prime Factorization:
- 12 = 2 x 2 x 3 = 22 x 3
- 18 = 2 x 3 x 3 = 2 x 32
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Identify All Prime Factors: The prime factors that appear are 2 and 3.
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Highest Power:
- The highest power of 2 is 22 (from the prime factorization of 12).
- The highest power of 3 is 32 (from the prime factorization of 18).
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Multiply: LCM = 22 x 32 = 4 x 9 = 36
Therefore, the LCM of 12 and 18 is 36.
Table Example: LCM of 15 and 18
| Number | Prime Factorization |
|---|---|
| 15 | 3 x 5 |
| 18 | 2 x 3 x 3 = 2 x 32 |
- Prime factors involved: 2, 3, 5
- Highest powers: 21, 32, 51
- LCM = 2 x 32 x 5 = 2 x 9 x 5 = 90
Key Takeaways:
- Prime Factorization is Key: The ability to accurately find the prime factorization of numbers is crucial.
- Consider All Factors: Don't forget any prime factors that appear in any of the numbers.
- Highest Power Matters: Always use the highest power of each prime factor.
By following these steps, you can effectively find the LCM of any set of numbers using prime factorization.
