The least common multiple (LCM) of 6 and 9 is 18.
Understanding the Least Common Multiple (LCM)
The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of all the given numbers. It is a fundamental concept in mathematics, particularly useful when working with fractions to find a common denominator.
How to Find the LCM of 6 and 9
To determine the LCM of 6 and 9, a common and straightforward method involves listing the multiples of each number until the first common multiple is identified.
Listing Multiples Method
According to educational resources, finding the LCM means identifying the smallest number that is exactly divisible by both numbers without leaving a remainder. Here's a step-by-step breakdown:
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List Multiples of 6: These are the numbers obtained by multiplying 6 by successive positive integers.
- 6 x 1 = 6
- 6 x 2 = 12
- 6 x 3 = 18
- 6 x 4 = 24
- ...and so on.
-
List Multiples of 9: Similarly, these are the results of multiplying 9 by successive positive integers.
- 9 x 1 = 9
- 9 x 2 = 18
- 9 x 3 = 27
- 9 x 4 = 36
- ...and so on.
To make it easier to compare, here's a table of the initial multiples for both numbers:
| Multiples of 6 | Multiples of 9 |
|---|---|
| 6 | 9 |
| 12 | 18 |
| 18 | 27 |
| 24 | 36 |
| 30 | 45 |
Why is 18 the LCM?
By examining the lists of multiples for both 6 and 9, the first (and smallest) number that appears in both lists is 18. This indicates that 18 is the smallest positive integer that can be evenly divided by both 6 and 9. Therefore, 18 is the least common multiple of 6 and 9.
Other Methods to Find LCM
While listing multiples is effective for smaller numbers, for larger numbers or multiple numbers, other methods such as prime factorization can be more efficient. The prime factorization method involves breaking down each number into its prime factors and then multiplying the highest power of each unique prime factor found.
