To find the Least Common Multiple (LCM) using the division method, you repeatedly divide the given numbers by prime numbers until you reach a row of numbers that are all 1 or are prime numbers themselves. Then, you multiply all the divisors (prime numbers) used to get the LCM.
Here's a detailed breakdown of the process:
Steps for Finding the LCM Using the Division Method
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Write the numbers: Arrange the numbers for which you want to find the LCM in a row, separated by commas.
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Divide by a prime number: Find the smallest prime number that divides at least two of the numbers in the row. Divide those numbers by the prime number. If a number is not divisible, simply bring it down to the next row.
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Repeat: Continue dividing the numbers in each subsequent row by prime numbers. Always start with the smallest possible prime number (2, 3, 5, 7, 11, etc.).
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Stop: Keep dividing until you reach a row where all the numbers are either 1 or are prime numbers themselves.
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Multiply: The LCM is the product of all the prime numbers you used as divisors.
Example
Let's find the LCM of 12, 18, and 30.
| Prime Divisor | 12 | 18 | 30 |
|---|---|---|---|
| 2 | 6 | 9 | 15 |
| 2 | 3 | 9 | 15 |
| 3 | 1 | 3 | 5 |
| 3 | 1 | 1 | 5 |
| 5 | 1 | 1 | 1 |
LCM = 2 x 2 x 3 x 3 x 5 = 180
Therefore, the LCM of 12, 18, and 30 is 180.
Key Considerations
- Prime Numbers: Always use prime numbers as divisors.
- Smallest First: Begin with the smallest prime number possible (2) and work your way up.
- Divisibility: If a number is not divisible by the prime number, simply carry it down to the next row.
- Efficiency: This method works efficiently for finding the LCM of multiple numbers.
