Least Common Multiple Problems

Are you struggling with the Least Common Multiple problems and finding no way to solve them? Have patience. Let’s help you solve the problems on LCM quickly. 

In this blog, you will learn about LCM (lowest common multiple) and will get tips to solve the Lowest Common Multiple problems using different methods. 

Let’s learn what LCM is.

Lowest Common Multiple (LCM)  

LCM, or the least common multiple of two or more than two numbers, is the least single value that is common among all multiples of the given numbers and divisible by all the numbers. 

We will understand it with the help of examples.

Example:

Let us consider two numbers, 3 and 5.

First, we will see the multiples of both numbers and then find the LCM.

Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30………

Multiples of 5 are 5, 10, 15, 20, 25, 30, 35……..

Now, see which number is common in the multiples of both. We can see that 15 and 30 is common in both multiples.

15 is less than 30. Therefore, 15 is the lowest common multiple of 15 and 30. 

We can write it as LCM (3,5)= 15

Methods to Find LCM 

Three different methods are used to find the LCM of numbers, including the Listing Method, the Prime Factorization Method, and the Division Method. We will learn each method in detail with examples.

LCM by Listing Method 

In the listing method of LCM, we calculate the lowest Common multiple by listing the multiples of numbers and picking the least common number.

Let’s understand it with an example.   

Consider we have two numbers, 2 and 9. We will find LCM of 2, 9 using the listing method. 

First, we will find the multiples of 2 and 9

• Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20…….. 
• Multiples of 9: 9, 18, 27, 36, 45, ………

Now, choose the smallest value that is common in both multiples. 

We can see that 18 is the smallest common number. 

LCM by Division Method 

The division method is the simplest and most commonly used method of calculating the lowest common multiple.  In this method, we divide the numbers by the smallest prime number and move on dividing until we get one at the end. 

Let’s understand it with the help of an example. 

Example: 

Consider we have two numbers, 6 and 20. We will find the LCM using the division method. 

1. We will choose a prime number that is a multiple of both or one number. 
2. 2 is the least prime number that is a factor of 6 and 20. Divide 6 and 20 by 2. On division, we are left with 3 and 10.
3. Divide the numbers again by 2 because 10 is still divisible by 2, and write three below as it is.
4. Now divide the numbers by 3. We are left with 1 and 5.
5. Divide by 5. Now, we have one on both columns. 
6. Multiply the numbers on the left. 

LCM by Prime Factorization Method 

Prime factorization is a method of calculating the least common multiple of numbers by finding the prime factors of numbers and writing them in exponential form. On multiplication, we can achieve the LCM of desired numbers. 

Let’s understand it through example. 

Example:

Suppose we have two numbers, 80 and 120. To find the LCM, we will follow these steps.

1. We will find the prime factors of the 80 and 120 by the repeated division method.

1. The prime factor of 80 is

The prime factor of 120 is

1. Find the LCM of 1200 and 800 by the prime factorization method.  

• To find the lowest common multiple of 16 and 20, we will find the prime factor of 1200 and 800 separately by the repeated division method.

• Now, write the factors in exponential form.