Least Common Multiple
The smallest multiple (other than zero) that two or more numbers have in common.
The Least Common Multiple (LCM) of any two or more integers is the smallest positive number that is a multiple of all of them. It can be calculated using a few different methods.
Method 1: Listing Multiples
One method for finding the LCM of two numbers is to list out their multiples until you find the smallest one they have in common. For example, to find the LCM of 6 and 8:
– Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …
– Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …
The smallest multiple they have in common is 24, so the LCM of 6 and 8 is 24.
Method 2: Prime Factorization
Another method for finding the LCM of two or more numbers is to use prime factorization. To use this method, find the prime factors of each number and then take the highest power of each factor that appears in any of the numbers. For example, to find the LCM of 12 and 18:
– Prime factors of 12: 2, 2, 3
– Prime factors of 18: 2, 3, 3
The highest power of 2 that appears is 2^2, and the highest power of 3 that appears is 3^2, so the LCM of 12 and 18 is 2^2 * 3^2 = 36.
Method 3: Using the GCD
A third method for finding the LCM of two numbers is to use the fact that the LCM * GCD = product of the two numbers. So if you know the GCD of two numbers, you can find their LCM by dividing their product by their GCD. For example, to find the LCM of 20 and 28:
– GCD of 20 and 28: 4 (you can use the Euclidean algorithm to find this)
– Product of 20 and 28: 560
So LCM of 20 and 28 = 560/4 = 140.
These are the three most commonly used methods to find the least common multiple of given numbers.
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