The Least Common Multiple (LCM) is the smallest positive integer that is divisible by two or more given integers. It represents the least common multiple of those integers, meaning it is the smallest number that can be evenly divided by all of them.
To find the LCM, there are a few different methods. One method is to list the multiples of each given number and find the smallest multiple that appears in all the lists. Another method involves the prime factorization of each number, where the LCM is found by multiplying the highest powers of all prime factors that appear in the factorizations.
For example, let's find the LCM of 4 and 6.
Method 1: Listing Multiples
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, ...
Multiples of 6: 6, 12, 18, 24, 30, 36, ...
From the lists above, we see that 12 is the smallest multiple that appears in both lists, so the LCM of 4 and 6 is 12.
Method 2: Prime Factorization
Prime factorization of 4: 2 * 2
Prime factorization of 6: 2 * 3
To find the LCM, we multiply the highest powers of all prime factors. In this case, the highest power of 2 is 2 and the highest power of 3 is 1, so the LCM of 4 and 6 is 2 * 2 * 3 = 12.
So, in either method, we find that the LCM of 4 and 6 is 12.
Another example:
Let's find the LCM of 10 and 15.
Method 1: Listing Multiples
Multiples of 10: 10, 20, 30, 40, 50, ...
Multiples of 15: 15, 30, 45, 60, ...
From the lists above, we see that 30 is the smallest multiple that appears in both lists, so the LCM of 10 and 15 is 30.
Method 2: Prime Factorization
Prime factorization of 10: 2 * 5
Prime factorization of 15: 3 * 5
To find the LCM, we multiply the highest powers of all prime factors. In this case, the highest power of 2 is 1, the highest power of 3 is 1, and the highest power of 5 is 1, so the LCM of 10 and 15 is 2 * 3 * 5 = 30.
So, in either method, we find that the LCM of 10 and 15 is 30.