# Least Common Multiple: An important Concept of Arithmetic Theory

The mathematical foundation is full of logic and concepts. This is really scary at times for the students. Cuemath describes the concepts in an efficient manner accompanied by examples making them easy to understand. Among such good mathematical topics, H.C.F and L.C.M form the base of many concepts at the concierge. H.C.Fand LCM are important to school mathematical topics that relate to the topic of factorisation or factors. In this article, the topic of LCM will be discussed. So, let’s get started.

## What is meant by factors?

When the numbers are divisible by other numbers in such a way that 0 is obtained as the remainder, the divisor is known as the factor of the dividend. For example,We know that 22 can be completely divided by 1, 2, 11 & 22. Then these numbers are the factors of 22.

## What is the meaning and definition of LCM?

The abbreviation of LCM: Lowest Common Factor.

The LCM can be defined as the minimum number that is a multiple of every number in any group of numbers. Simply, the lowest value that is divisible by each member of any group of numbers is the LCM for that group.

For example: What is the LCM of 5 & 3?

Solution: The multiples of 5 are 5, 10, 15, 20, ……and so on.

Similarly, the multiples of 3 are 3, 6, 9, 12, 15, 18……. and so on.

Now, we noticed that 15 is the smallest number that is common there so, the LCM of 5 & 3 is 15.

## Relation between HCF and LCM

Product of both i.e., HCF * LCM= Product of the respective numbers.

Assuming that the two numbers arex, y then, according to the above reference

(HCF of x, y) *(LCM of x, y) = x* y

This implies, HCF of two numbers can be given by the relation, HCF= product/ LCM

And LCM= product/ HCF

## Sometips and tricks for LCM

• The LCM of any group of numbers is always equal or greater than the numbers but never less than any of them. In other words, the LCM of the set of numbers is the smallest integer that is divisible by each of the members of the respective set.
• The product of the prime numbers is taken as the LCM of the same group of prime numbers.
• The product of the two numbers is the same as the product of the HCF and LCM of the respective numbers.
• The LCM in the case of fractions is calculated by the relation, LCM of the numerator divided by HCF of the denominator.
• For finding the LCM of any given set of numbers any of the prime factorisation or long division methods can be used.

## Calculations of LCM according to different methods

1. Prime factorisation method: In this method, the prime factors of each and every member of the given set of numbers are considered or used to calculate the LCM of the given set.
• Example: Calculate the lowest common multiple of 3 & 33.

Solution: We know that 3 be written as 3*1

While33 can be represented by 1*3*11

So, the LCM would be 1*3*11= 33

1. Division method or long division method: This technique is the same as we do for HCF. The only difference is that for HCF we take or factories each member of the set individually whereas contradictory to this for LCM we take all the numbers simultaneously in one solution and multiply the numbers obtained at the end of the division.This is also sometimes listed as a common factor grid.

## Solved example

1. 5, 7, 9, 12 is a set of numbers whose members divide a certain number leaving 3 as the remainder in each case. What is the number divided by the members ofthe set?

Here, the LCM of the divisors when added with the common difference will give the respective dividend.

So, the LCM of 5, 7, 9, 12 is 3*5*7*3*4=1260

Hence the required integer will be 1260+3= 1263.