LCM Formula

LCM Formula allows students to find out the smallest common multiple that is divisible by given set of numbers. LCM Formula is essential to learnh to solve basic problems such as addition and subtraction of fraction, rationalization of denominators. Learn LCM formula and how to solve examples based on them in this article by Extramarks

What is LCM?

The LCM, or Least Common Multiple, of two or more numbers is the smallest multiple that can be divided by each of the provided numbers without leaving any remainder. In other words, it is the lowest positive integer that is a multiple of all of the numbers provided.

Let’s say we have to find the LCM of 6 and 8.

• Multiples of 6: 6, 12, 18, 24, 30, …
• Multiples of 8: 8, 16, 24, 32, 40, …
• The first common multiple of 6 and 8 is 24.
• So, the LCM of 6 and 8 is 24.

In this case, the LCM of 6 and 8 is 24, which is the lowest multiple that divides both 6 and 8 without leaving a remainder. It’s worth noting that there may be other common multiples of 6 and 8 (e.g., 48, 72), but the LCM is the smallest.

What is LCM Formula?

The Least Common Multiple (LCM) refers to the common multiple that is the least (smallest) among all multiples for the given set of numbers. There are basically two LCM Formulas mentioned below:

• LCM of a, b = (a x b)/HCF(a, b)
• LCM of (a/b) and (c/d) = LCM of Numerators/HCF of Denominators = LCM of (a, c)/HCF of (c, d)

LCM Formula for Two Numbers

The lcm formula for two numbers is given as ratio of product of two numbers and HCF of given numbers. It is expressed as

LCM of a, b = (a x b)/HCF(a, b)

LCM Formula for Two Fractions

The lcm formula for two fractions is given as ratio of LCM of numerators and HCF of denomiantors. It is expressed as

LCM of (a/b) and (c/d) = LCM of (a, c)/HCF of (c, d)

How to Find LCM?

There are three methods to find LCM

• Listing Multiple Method
• Prime Factorization Method
• Long Divison Method

LCM using Listing Multiples Method

In this method, we basically list out all the first few multiples and the smallest  common multiple is the LCM of the numbers.

LCM Using Prime Factorization Method

In this method, we list out all the prime factors and find out the common and uncommon factors. Multiply the common factors taken maximum number of times and all the uncommon factors to find the LCM

LCM using Division Method

In LCM using division method we divide the numbers with smallest common factors, the quotient so obtained is again divided. This process continue till the quotiet becomes one. Now multiply all the divisors to obtain the LCM

Let’s understand all these methods using solved examples below

Examples Using Least Common Multiple Formula

Example 1: Find LCM of 12 and 16 using Listing Multiple Method

Solution:

Multiple of 12 = 12, 24, 36, 48, 60…

Multiples of 16 = 16, 32, 48, 64, 80…

We see that the smallst common multiple is 48. Hence, multiple of 12 and 16 is 48

Example 2: Find LCM of 12 and 18 using Prime Factorization Method?

Solution:

Prime factors of 12 and 18 are

12 = 2 × 2 × 3

18 = 2 × 3 × 3

Common factors: 2 (included twice), 3 (included twice)

LCM = 2 × 2 × 3 × 3 = 36

Example 3: Given that the HCF of two numbers a and b is 6, and their product a×b is 72, find their LCM.

Solution:

Using the formula:
LCM(a,b)×HCF(a,b)=∣a×b∣

We substitute the given values:
LCM(a,b)×6=72

Now, we solve for
LCM(a,b)=
6/72 =12

So, the LCM of the two numbers is 12.

Example 4: Find the LCM of 2/3 and 4/9

Solution:

LCM of 2/3 and 4/9 = LCM of (2, 4)/ HCF of (3, 9) = 2/3