# Playing With Numbers

#### Lowest Common Multiple(LCM)

Lowest Common Multiple. The Lowest Common Multiple (or LCM) of two or more given numbers is the lowest (or smallest or least) of their common multiples.

Methods to find the Least Common Multiple (LCM)

We can calculate the LCM of few given numbers in the following ways.

Method 1 – LCM by Listing Multiples

It consists of the following steps.

• Step 1. List the multiples of each number until at least one of the multiples appears on all lists.
• Step 2. Find the smallest number that is on all of the lists
• Step 3. This number is the LCM

Example: Find the LCM of 6,7 and 21.

Solution. Following the above steps, we have

• Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, . . .
• Multiples of 7: 7, 14, 21, 28, 35, 42, 56, 63, 70, . . .
• Multiples of 21: 21, 42, 63, 84, . . .

The smallest number that is there in all the lists is 42. Therefore, LCM(6, 7, 21) is 42.

Method 2 – LCM by Prime Factorisation

It consists of the following steps.

• Step 1. Find all the prime factors of each given number.
• Step 2. List all the prime numbers found, as many times as they occur most often for any one given number.
• Step 3. Multiply the list of prime factors together to find the LCM.

The process is valid for finding the LCM of any number of numbers.

Example. Find the LCM of 24 and 90.

Solution. The prime factorisations of 24 and 90 are:

• 24 = 2 × 2 × 2 × 3;
• 90 = 2 × 3 × 3 × 5.

From the above prime factorisations, note that

• the prime factor 2 appears maximum number of three times in the prime factorisation of 24;
• the prime factor 3 occurs maximum number of two times in the prime factorisation of 90;
• the prime factor 5 occurs only once in 90.

Therefore, LCM (24, 90) = (2 × 2 × 2) × (3 × 3) × 5 = 360

Example. Find the LCM of 40, 48 and 45.

Solution. The prime factorisations of 40, 48 and 45 are:

• 40 = 2 × 2 × 2 × 5
• 48 = 2 × 2 × 2 × 2 × 3
• 45 = 3 × 3 × 5

From the above prime factorisations, note that

• the prime factor 2 appears maximum number of four times in the prime factorisation of 48;
• the prime factor 3 occurs maximum number of two times in the prime factorisation of 45;
• the prime factor 5 appears one time in the prime factorisations of 40 and 45.

Therefore, LCM (24, 90) = (2 × 2 × 2) × (3 × 3) × 5 = 360.

Method 3 – LCM by Division Method

It consists of the following steps.

• Step 1. Write the given numbers in a row (maintaining a sufficient distance between them);
• Step 2. Starting with the lowest prime numbers, divide the row of numbers by a prime number that is evenly divisible into at least one of the your number and bring down the result into the next table row.
• Step 3. If any number in the row is not evenly divisible just bring down that number.
• Step 4. Continue dividing rows by prime numbers that divide evenly into at least one number.
• Step 5. Stop when the last row of result is all 1’s.

The process is valid for finding the LCM of any number of numbers.

Example. Find the LCM of 20, 25 and 30.

Solution. Following the above steps, we have

Therefore, LCM (20, 25, 30) = 2 × 2 × 3 × 5 × 5 = 300.

Step by Step Explanation

Divide the row of numbers by the least prime number which divides at least one of the given numbers. It is 2 here. The numbers like 25 are not divisible by 2 so they are written as such in the next row. The numbers that are not divisible by 2 (like 25) will come to the next row. Again divide by 2. This will continue till we have no multiples of 2.

Divide by 3, the next prime number. Continue till we have no multiples of 3.

Divide by 5, the next prime number. Continue till we have no multiples of 5.

We will stop as all the numbers are 1.

Method 4 – LCM by HCF

The key idea is to use the formula LCM(a, b) × HCF(a, b) = a × b for any two natural numbers a and b. We thus find

Example. Find LCM of 16 and 44.

Solution. We find that HCF (16, 44) = 4 and product of the numbers = 16 × 44 = 704.