LCM of 3,8, and 12

The common multiple of 3, 8, and 12 is 24  Here, we will learn about the LCM of 3 numbers. Children learn about LCM at younger ages. Here, we will discuss the methods used for finding out LCM.
 

Out of many methods, prime factorization method is widely used for its easy approach. Here, we will learn about other methods as well. A few commonly used methods are as follows - 

1. Listing Of Multiples
2. Prime Factorization
3. Division Method
    

Listing multiples can be a tedious method for finding the LCM. Here, the listing of multiples for all these 3 numbers is noted - 

• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24…
• Multiples of 8: 8, 16, 24…
• Multiples of 12: 12, 24…

Then we can see that out of 3, 8, and 12, 24 is the smallest common number that is present in them. So we see that 24 is the LCM of 3, 8, and 12.
 

The product of the highest power of prime factors of 3, 8, and 12 is the LCM of these numbers. So let us look at it step by step to understand it better.

Breaking the given numbers into their prime factors

Prime factorization of 3 = 3¹
Prime factorization of 8 =2³
Prime factorization of 12 = 2²  × 3¹ 

Multiplying the highest power of prime factors: 2³ × 3¹ → 8 × 3 = 24
LCM of 3, 8, and 12 is 24.

In this method, we will be dividing the given numbers with the common prime factors until all numbers are reduced to 1. Let us look at this step by step and make it easy for the children to learn it.

Step 1: Arrange the number in a sequence, divisors, and the numbers are on the left and right sides respectively.

Step 2: For finding the divisor, it is always the smallest common prime factor. Here, the smallest common prime factor is 2. Dividing 3, 8, and 12 by 2. The result is 3, 4, and 6. 

Step 3: As 4 and 6 are divisible by 2, again the divisor is 2. Dividing 3, 4, and 6 by 2. Now the remainder is 3, 2, and 3.

Step 4: Continue dividing the numbers with the smallest prime number until all numbers are reduced to 1.

The divisors are 2, 2, 2, 3. LCM of 3, 8, and 12 is the product of divisors.

Hence, the LCM of (3, 8, and 12) = 2 × 2 × 2 × 3 = 24
 

• Factor: A number that will divide two or more numbers, leaving no remainder. For 18 and 24 we have 6 as a common factor, it means both 18 and 24 can be divisible by 6.

• Prime Factorization: When a number can be represented as the factors of prime numbers, it is called prime factorization. The prime factorization of 18 for example is 2×3×3.

• Greatest Common Factor (GCF): GCF is the greatest factor that is common in the given numbers. For example, the GCF of 5, 10, and 15 is 5. Because the common factors of 5 and 10 are 1 and 5.

• Division Method: In the division method, the numbers are divided by the smallest common prime factor till the numbers are reduced to 1.