Contents
Least Common Multiple used, for example, to reduce common denominators to a common denominator, allowing you to add, subtract, or compare them.
In this article, we will understand how this operation is performed, as well as analyze practical examples to consolidate the material presented.
Finding the Least Common Multiple
Number b is multiple number a, if b completely (without remainder) is divided into a. Pronounced as “b multiples a“. Denoted by letter К.
Examples of multiples:
- multiples of 3 or K(3): 6, 9, 12,15, 18 etc.
- multiples of 7 or K(7): 14, 21, 28, 35, 42 etc.
There can be an infinite number of multiples.
Common multiple of two natural numbers is a number that is evenly divisible by both of these numbers.
Least common multiple of two natural numbers is the smallest number of common multiples of these numbers. Designated as NOK
For example, the NOK (5, 9) is the least common multiple of 5 and 9.
Finding the NOC
To find the least common multiple, you can use one of the two methods below:
For two/small numbers
When we are dealing with two numbers (or small ones), the process of finding the LCM consists of the following steps:
- We write down the multiples for each number in ascending order.
- We find the first match in the received series of numbers. This is the NOC.
Example
Let’s find the least common multiple of the numbers 6 and 14.
Solution
Multiples of 6: 12, 18, 24, 30, 36, 42, 48 etc.
Multiples of 14:28, 42, 56 etc.
So LCM(6, 14) = 42.
For multiple/large numbers
This method is suitable if we are dealing with large numbers, or when we need to find the LCM for three or more numbers.
- First, we decompose the numbers into prime factors – prime numbers that divide the number by an integer (their number for different numbers can also be different). For convenience, we start with the smallest value and end with the largest.
- Among the factors of the smaller number, we find the one that was not included in the larger one. We do the same with the next ascending number/numbers.
- We multiply the larger number by the found additional factors and get the LCM.
Example
Let’s find the LCM (12, 28, 32).
Solution
Let’s decompose these numbers into prime factors.
Among the factors of the smaller number (12), the number 32 is not included in the larger number (3), among the factors of the middle number (28), the number 7 is not included.
Therefore, LCM (12, 28, 32) = 32 ⋅ 3 ⋅ 7 = 672.
Other cases
1. If one of the numbers for which it is required to find the least common multiple is completely divisible by other numbers, then this number is the LCM.
For example: LCM (20, 40, 80) = 80.
2. LCM of relatively prime numbers is the product of these numbers, because they have no common prime factors.
For example: LCM (3, 5) = 3 ⋅ 5 = 15