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LCM of Two Numbers in Java

In arithmetic, the Least Common Multiple (LCM) of two or more numbers is the least positive number that can be divided by both the numbers, without leaving the remainder. It is also known as Lowest Common Multiple (LCM), Least Common Denominator, and Smallest Common Multiple. It is denoted by LCM (a, b) or lcm (a, b) where a and b are two integers.

It is used when we add, subtract, or compare the fractions. While we perform addition or subtraction of the fractions, we find the LCM of the denominators and then solve the fractions. The LCM of the denominators is known as the Least Common Denominator (LCD).

Properties of LCM

  • Associative: LCM (a, b) = LCM (b, a)
  • Commutative: LCM (a, b, c) = LCM (LCM (a, b), c) = LCM (a, LCM (b, c))
  • Distributive: LCM (ka, kb, kc) = kLCM (a, b, c)
  • The LCM is related to GCF:
LCM of Two Numbers in Java

How to Find LCM?

There are four methods to find the LCM, are as follows:

  • Using the Greatest Common Divisor (GCD) Method
  • Using Prime Factorization Method
  • Using Multiples of Numbers

Using the Greatest Common Divisor (GCD) Method

We can also calculate the LCM by using the GCD. The formula for LCM using the GCD is:

LCM of Two Numbers in Java

Greatest Common Divisor: It is the highest number that completely divides two or more numbers. It is abbreviated for GCD. It is also known as the Greatest Common Factor (GCF) or the Highest Common Factor (HCF).

How to Find GCD?

In order to find the GCD follow the steps given below:

  • Write all the factors of each number.
  • Select the common factors.
  • Select the largest number, as GCD.

Let's understand it through examples.

Example: Find the LCM of 8 and 10.

Solution:

According to the formula that we have learned above:

LCM of Two Numbers in Java

First, we find the GCD of 8 and 10.

Factors of 8: 1, 2, 4, 8

Factors of 10: 1, 2, 5, 10

Common Factors: 1, 2

Greatest Common Divisor: 2

LCM of Two Numbers in Java

Hence, the LCM of 8 and 10 is 40.

Let's implement the logic in a Java program.

Using if Statement

LcmExample1.java

Output:

The LCM of 12 and 9 is 36.

Using Recursion

LcmExample2.java

Output:

Enter the first number: 45
Enter the second number: 78
LCM of 45 and 78 is 1170

Using Multiples of Numbers

It is a very lengthy method, so it is not usually used. Follow the steps given below to find the LCM using multiples of numbers.

  • List all the multiple of each number until the first common multiple is found.
  • Pick the smallest multiple that is common in all the given numbers.

Let's understand it through examples.

Example: What is the LCM of 9 and 12.

Solution:

Multiples of 9: 9, 18, 27, 36

Multiples of 12: 12, 24, 36

LCM (9,12)=36 because it is a common multiple in both.

Hence, the LCM of 9 and 12 is 36.

Let's implement the logic in a Java program.

LcmExample4.java

Output:

Enter the first number: 123
Enter the second number: 213
LCM of 123 and 213 is: 8733






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