LCM Formula: Definitions, Methods and Examples

When the multiples of two or more numbers are the same, this is known as a common multiple. Their least common multiple (LCM) is the smallest and most common multiple for both numbers out of all possible multiples. To find the least common multiple of two numbers, use the LCM Formula.

LCM stands for the smallest positive integer divisible by both a and b (say a, b are two integers). Alternatively, the smallest positive number that is a multiple of two or more numbers is the LCM of two numbers.

Also Read: Relation Between HCF and LCM

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Least Common Multiple

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The smallest positive integer divisible by both a and b is the Least Common Multiple (LCM) of two integers a and b. As a result, the smallest positive number is a multiple of two or more. For example:

1. To calculate the LCM of (75, 55), we will find factors of 70 and 55, giving

75= 3 x 5 x 5.

55= 5 x 11.

Common prime factors are 3, 5, 5, 11.

Therefore, the least common multiple= 3 x 5 x 5 x 11 = 825.

2. To calculate the LCM of (15,25, 35), we will find the factors of 15, 25, and 35, as

15= 3 x 5

25 = 5 x 5

35 = 5 x 7

Common prime factors are 3, 5, 5, 7

Therefore, the least common multiple = 3 x 5 x 5 x7 = 525.

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Formula

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The LCM formula for any two numbers and fractions using GCD (HCF) is given below.

• If we know the greatest common divisor (GCD) of two numbers, then the Least Common Multiple can be easily calculated using the following least common multiple formula:

• In order to find the LCM of two fractions, the first step is to find the LCM of the numerator(s) and the HCF of the denominator(s). These outcomes will be then expressed as a fraction as well. Thus,

LCM=\(\frac{LCM of Numerator}{HCF of Denominator}\)

Quick Read: Euclid's Division Lemma

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Methods

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1. Prime Factorization for determining LCM.

Steps:

1. Find all the prime factors of the number.
2. List all prime numbers discovered, in the order in which they appear most frequently for each number.
3. To find the LCM, multiply all of the prime factors together.

Let a and b be two numbers, then the prime factorization of both a and b is used to compute the LCM(a,b).

For example, we find the following for LCM(24, 60):

 24 = 2 x 2 x 2 x 3.

 60 = 2 x 2 x 3 x 5 

Using all prime factors found in the order in which they occur most frequently, we get 2 x 2 x 2 x 3 x 5 = 120, and thus LCM(24, 60) = 120.

2. LCM using the Division Method:

1. First, separate the given numbers with commas in a horizontal line.
2. Then we must divide all the numbers given by the smallest prime number.
3. The quotients and undivided numbers must now be written in a new line beneath the previous one.
4. Repeat this process until there are no prime factors in common.

The product of all divisors and the numbers in the last line is LCM.

Also Check: Real Numbers Formula

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Things to Remember

• The product of two numbers' L.C.M. and H.C.F. is the same as the product of the numbers. The L.C.M. of 6 and 12 is 12, and the H.C.F. of 6 and 12 is 6. We see that the product of 6 and 12 is also the product of 6 and 12 L.C.M. and H.C.F.
• Properties of L.C.M: 
  • L.C.M is associative.
  • L.C.M is commutative.
  • L.C.M is distributive.
• Greatest Common Factor (GCF): It's important to remember what a factor of a number is. A factor is a number that evenly divides another number.
  • Here, 1,2,3,4,6,8, 12, and 24 are all possible divisions of 24. As a result, 1,2,3,6, 8,12, and 24 are all factors of 24.
  • 1,2,3,4,6,9,12,18, and 36 are all possible divisions of 36. As a result, factors of 36 are 1,2,3,4,6,9,12, and 36.
  • As a result, the greatest common factor of 24 and 36 is 12.
• The LCM (Lowest Common Multiple) of two or more numbers is the smallest of their common multiples. Several methods exist for determining the LCM of two or more numbers. LCM is defined to be zero if either a or b is zero.

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