LCM of 8, 12 and 18

The number that is the least common multiple of 8, 12, and 18 is 72. This number is the smallest number with a positive value divisible by 8, 12, and 18. The LCM of any number will always be positive, as a negative value does not exist based on the concept of LCM. In the case of two irrational numbers, the LCM does not exist. 

There are multiple methods to find the LCM of 8, 12, and 18. The methods are listed below:

• Listing of Multiples
• Prime Factorization
• Division Method
• Cake Method/ Ladder Method

This method is one of the simplest methods to find the LCM of any given number. In this method, the multiples of each number are listed. The multiples are then compared together to find the common multiple that comes in the list of multiples of all three numbers. The multiples of 8, 12, and 18 are identified and then the common number present in all three lists is the LCM.

Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88…

Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96…

Multiples of 18 = 18, 36, 54, 72, 90, 108…

72 is the first number that comes out in all three lists of multiples. Hence, 72 is the LCM of 8, 12, and 18.

Here, the prime factorization of the numbers is done to find their LCM. Factorization is done using prime numbers until the number is completely broken down. The common numbers that come in the factorization of all three numbers are taken as a single factor. The multiplication of the prime factors is calculated, and the product obtained will be the LCM.

To find the LCM of 8, 12, and 18, the prime factorization of each number is to be done, and then prime factors are to be multiplied together.

Prime factorization of 8 = 2 x 2 x 2 = 23

Prime factorization of 12 = 2 × 2 × 3 = 22 x 31

Prime factorization of 18 = 2 × 3 × 3 = 21 × 32

The LCM of 8, 12, and 18 = 2 × 2 × 2 × 3 × 3 = 23 × 32 = 72.

The division method involves the division of numbers together in an order. The divisor is to be selected in such a way that the number is able to divide all the numbers and is a prime number. If the number is not divisible,

the number is carried down. Another number is selected by which the division is possible. The division is continued till the remainder is 1 for all the numbers. The divisors with which the division is done are taken together and multiplied to find the LCM of the numbers.

While applying the division method to find the LCM of 8, 12, and 18 with prime numbers till the number becomes completely divided and 1 remains as the remainder. 

Step 1: The numbers 8, 12, and 18 are to be divided together by prime number 2.

Step 2: Division is continued again by 2, and we get 4, 6, and 9.

Step 3: The same step is again repeated by continuing division by 2 we get 2, 3, and 9.

Step 4: Again 2 is taken up for division, and we get 1, 3, and 9.

Step 5: Division is repeated with 3 and the remainders are 1, 1, and 3.

Step 6: 3 is used to continue the division and the remainder are 1, 1, and 1.

The divisors are 2 × 2 × 2 × 3 × 3 = 72, Thus LCM is 72.
 

• Prime Factorization: The process of breaking down the numbers by using the prime factors of that particular number.

• Multiples: The numbers that are the result of multiplying two numbers with each other.

• GCD: Greatest Common Divisor or the largest number that is an integer and has the ability to divide two or more numbers.

• Positive Integers: The whole numbers that more than 0 and extends till infinity. They can be denoted as Z+ in mathematics.

• Prime numbers: The numbers that can only be divided by two numbers 1 and itself and not by any other numbers.

• Prime Factors: The factors of a number that are included in the list of prime numbers.