LCM Full Form

The LCM full form is Least Common Multiple of a few natural numbers is the smallest of the natural numbers that are divided with all of these numbers. There lies no greatest total multiple as the sequence of common multiples tends to infinity. To find the LCM of any number, the decomposition of numbers into simple multipliers can be done, but in actual practice, when the numbers are small, the smallest total multiple is easier to pick up.

What is the full form of LCM?

LCM full form is Least Common Multiple. The minimum common multiple of two numbers taken as ‘a’ and ‘b’ (Least Common Multiple) is the minimum multiple that is divided by both ‘a’ and ‘b’ and is recorded as LCM representation (a, b). The LCM of two or more numbers is considered as the smallest positive integer that is divisible by all the given numbers.

The minimum common multiple of more than two integers or rational numbers is clearly stated: it is the smallest number and there are integer multiples of each. While calculating the Minimum Common Multiple or Least Common Multiple (LCM), the maximum number of conventions is generally used to aid the calculation.

The Lowest Common Multiple (LCM) is the other term used as a substitute for the Least Common Multiple (LCM). The smallest number by which all the numbers such as d, e, f, etc are divisible is called the LCM of two or more numbers, such as d, e, f, etc. Moreover, L.C.M. is the smallest number consisting of the elements d,e,f etc. 

For example, if you consider the integers 3 and 5, then the least common multiple (LCM) of 3 and 5 is 15, which is the smallest number by which 3 and 5 are divisible respectively.

Importance of Learning LCM

The concept of HCF and LCM occupy unimportant places both in school-level mathematics as well as in higher classes. Basic Arithmetic plays a crucial role when it comes to building and creating a solid foundation in Mathematics. But, Mathematics is both difficult and interesting at the same time. If the concepts are clear right from the very beginning, then students will find it interesting to solve and they will not face challenges in understanding the advanced concepts that are taught in the higher classes. Students must understand all the important concepts of LCM and HCF problems, definitions, formulas and methods of solving along with LCM and HCF questions with examples. Key to grasping knowledge lies in solving as many questions as possible.

What is HCF?

The full form of HCF is known as the Highest Common Factor. HCF is also called GCF whereas GCF’s full form is Greatest Common Factor. The LCM full form is the Lowest Common Multiple. The Greatest Common Factor (GCF or GCD or HCF) for a subset of whole numbers is the largest positive integer that divides all the given numbers obtaining zero as its remainder.

What is a Factor?

Factors of a number are the numbers that divide the given number exactly with 0. When dividing the numbers with 0, factors of a number are the exact divisors of the given numbers. However, the factor of a number cannot be a fraction. Therefore, a factor of a number is the divisor of a provided number.  The easiest method of finding the factors of a number is to divide it by the smallest prime number.

For example, considering the number 15, it is divisible by 1,3 and 5 respectively. 

Methods to Find Least Common Multiple (LCM)

There are various methods to calculate and find the LCM of various numbers. Different methods to determine the LCM of a set of numbers are mentioned below:

• Listing the multiples
• Prime Factorization Method
• Division Method

Listing the Multiples Method

To find the LCM using the multiples method, list the multiples of the numbers. The least common multiple is the first common multiple for the given numbers.

So, for 12 and 16, the number 48 is the LCM.

Prime Factorization Method

The prime factorization method involves finding the prime factors of the provided numbers and identifying the least common multiple (LCM). 

For example, the prime factorization of 40 can be done in the following way:

Prime factorization of 40= 2*2*2*5

= 8*5

=40

Division Method

In the division method, the numbers given are divided by the common divisors until there is no possible further division by the same number. Further, the divisors and the remainders are multiplied to get the least common multiple.

What is the Relation Between LCM and HCF?

The highest common factor of two or more numbers is known as the HCF of the given numbers. Whereas, the smallest number among all common multiples of the given numbers is considered as the least common multiple of two or more numbers. 

For example, assuming a and b are two numbers, then the formula for determining the relationship between their LCM and HCF is:

LCM (a, b) × HCF (a, b) = a × b

Various examples on Finding LCM and HCF

Example 1:

Find the Highest Common Factor of 25, 35 and 45.

Solution:

Given, three numbers are 25, 35 and 45

As we know, 25 = 5 × 5

35 = 5 × 7

45 = 5 × 9

From the above expression, it can be said that 5 is the only common factor for all three numbers.

Therefore, 5 will be considered as the HCF of 25, 35 and 45.

Example 2:

Find the Least Common Multiple of 36 and 44.

Solution: 

Given, two numbers 36 and 44. We can find out the LCM by division method by:

Therefore, the LCM of 36 and 44 = 2 × 2 × 3 × 3 × 11 = 396

Example 3:

What is the L.C.M. of 25, 30, 35 and 40?

Solution:

L.C.M. of 25, 30, 35 and 40

We can find LCM by prime factorisation as:

Prime factorisation of 25 = 5 x 5 = 52

Prime factorisation of 30 = 2 x 3 x 5 

Prime factorisation of 35 = 5 x 7

Prime factorisation of 40 = 2 x 2 x 2 x 5 = 23 x 5

Thus, 

LCM of 25, 30, 35 and 40 = 2 x 2 x 2 x 3 x 5 x 5 x 7 = 4200

Example 4:

The HCF of the two numbers is 29 & their sum is 174. What are the numbers?

Solution:

Let us consider the two numbers to be 29x and 29y. 

Given, 29x + 29y = 174 

29(x + y) = 174 

x + y = 174/29 = 6

Since x and y are co-primes, therefore, possible combinations would be (1,5), (2,4), (3,3).

The only combination that follows the co-prime part is (1,5)

For (1,5): 29 x = 29 x 1 and 29 y = 29 (5) = 145

Therefore, the required numbers are 29 and 145.

Some of the interesting tricks to find the HCF and LCM are given below:

• The HCF of given numbers is never greater than any of the numbers.
• The LCM of given numbers is never less than any of the numbers.
• The LCM of two or more prime numbers is nothing but their product.

What is the Formula for HCF and LCM?

Product of Two numbers = (HCF of the two numbers) x (LCM of the two numbers)

Hence, HCF of Two numbers = Product of Two numbers/L.C.M of two numbers

LCM of two numbers = Product of Two numbers/H.C.F. of Two numbers

We can find LCM and HCF using the prime factorization method and long division method