The least common multiple of a set of numbers such as a and b is the smallest positive integer that is divisible by both a and b. Least Common Multiple is often referred to as LCM.
Example: What are the common multiples of 3 and 5?
Both 3 and 5 have an infinite number of multiples, therefore they also have an infinite number of common multiples.
The first 5 multiples of 3 are: 3, 6, 9, 12, 15
The first 5 multiples of 5 are: 5, 10, 15, 20, 25
The following are the first 5 common multiples of 3 and 5. In other words, the multiples of each that they have in common: 15, 30, 45, 60, 75
To find the next common multiple we add 15 to the previous multiple in the list. Therefore, we can say that the common multiples of both 3 and 5 can be expressed as 15m, where m is an integer.
So what is the Least Common Multiple? Well, it’s just the smallest common multiple between the set of numbers. In the case of 3 and 5, the LCM is 15.
The method described above for finding the LCM of a set of numbers works but can quickly become very tedious and inefficient.
There is a better way of finding the LCM of any set of numbers.
- First, we list the prime factors of each number
- Then we multiply the prime factor that occurs the greatest amount of times in any of the numbers
- Finally, we check that all numbers evenly divide our result.
Example: What is the Least Common Multiple (LCM) of 8, 105, and 72?
- 8 = 23, 105 = 3 * 5 * 7, 72 = 23 * 32
- 2 occurs the most 3 times so we take 23. 3 occurs the most 2 times so we take 32. 5 and 7 occur the most 1 time so we take 5, and 7. All together we end up with 23, 32, 5, and 7. Multiplying these altogether we get 23 * 32 * 5 * 7 = 2520. LCM(8, 105, 72) = 2520
- 2520/8 = 315, 2520/105 = 24, 2520/72 = 35