What is LCM?
The least common multiple or LCM is the smallest positive integer that is divisible by the given integers. We can list out the multiples of the given numbers to find the LCM. This means that the LCM has to be a multiple of all the common numbers. Thus, in other words, we can say that the LCM of the given numbers is the smallest value that can appear as a multiple, which can be divided by any of the given numbers. The least common multiple of r and q can be represented by LCM (r, q). When we need to perform simple arithmetic operations on unlike fractions such as addition, subtraction, or comparison, we need to use the LCM.
What are Prime Numbers?
A prime number can be defined as a number that is greater than 1, but it cannot be expressed as a product of two more whole numbers. The only factors of a prime number are the number itself and 1. On the other hand, a composite number can be defined as a number greater than one, which is not a prime number. The Fundamental Theorem of Arithmetic is based on prime numbers. According to this theorem, any number greater than 1 is either a prime or can be expressed as a product of primes. This product is always unique to a particular number. For example, if we consider a composite number 20. It can be broken into prime factors given by 2 * 2 * 5. If 1 were included in the list of prime numbers, then the prime factorization of numbers would not have been unique thus, rendering the theorem null and void.
Prime Factorization for LCM Calculation
There are several methods that can be used to find the LCM of given numbers. Usually, we list the multiples of each number and write down all the common multiples. The multiple with the smallest value gives the LCM. This is called the listing method and works well for small numbers. We know that the multiples of a number that can be listed are endless. Hence, when we have to work with numbers that have four digits or more, the process of finding the LCM can be very time-consuming. To optimize this process, we turn to the prime factorization method. We first list down the prime factors of all the given numbers. Next, we write down all the prime factors ensuring that the common ones have been noted only once. To find the LCM, we have to multiply all the prime factors. Given below is an example to help you to understand the concept.
Find the LCM of 4 and 18.
Step 1: Express the given numbers as a product of their prime factors
- Prime Factors of 4: 2 * 2
- Prime Factors of 18: 2 * 3 * 3
Step 2: Write down all factors without repetition
- Factors : 2 * 2 * 3 * 3
Step 3: Multiply all factors
- LCM (4, 18) = 36
Thus, prime numbers can be used to calculate the LCM of numbers quickly and effectively.
