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LCM of 12, 15 and 21: How to find LCM of 12, 15 and 21?

The LCM of 12, 15 and 21 is 420. Least Common Multiple (LCM) of a set of numbers is the smallest number that all of them can multiply into. In this blog, we’ll explain simple ways to find the LCM of 12, 15, and 21.

Methods to find the LCM of 12, 15, and 21

1. Prime Factorization Method

Break each number into its prime factors and use the highest power of each factor.

  • 12 = 2 × 2 × 3 (or 2² × 3)
  • 15 = 3 × 5
  • 21 = 3 × 7
  • Taking the highest powers: 2² × 3 × 5 × 7 = 420

Thus, LCM(12, 15, 21) = 420.

2. Listing Multiples Method

List the multiples of each number and find the smallest one they all share.

  • Multiples of 12: 12, 24, 36, 48, 60, ..., 420, ...
  • Multiples of 15: 15, 30, 45, 60, ..., 420, ...
  • Multiples of 21: 21, 42, 63, ..., 420, ...
  • The smallest common multiple is 420.

Thus, LCM(12, 15, 21) = 420.

3. Division Method

Divide all numbers by common factors until at least 2 numbers are divisible.

  1. Divide 12, 15, and 21 by 3 → (4, 5, 7)
  2. 4, 5 and 7 all are divisible by 1 only
  3. Multiply the divisors: 3 × 4 × 5 × 7 = 420

Thus, LCM(12, 15, 21) = 420.

Know how to find the lcm of 48, 72 and 92 here.

Conclusion:

The LCM of 12, 15, and 21 is 420, and you can find it using prime factorization, listing multiples, or division.

FAQs

1. What is the LCM of 12, 15, and 21?

The LCM of 12, 15, and 21 is 420. This is the smallest number that is a multiple of all three numbers. ​

2. How to find the LCM of 12, 15, and 21?

There are several methods to determine the LCM:​

  • Prime Factorization Method: Break down each number into its prime factors:

    • 12 = 2² × 3

    • 15 = 3 × 5

    • 21 = 3 × 7​

  • The LCM is the product of the highest powers of all prime factors:

    • LCM = 2² × 3 × 5 × 7 = 420​

  • Division Method: Divide the numbers by common prime factors and multiply the divisors to get the LCM. ​

  • Listing Multiples Method: List the multiples of each number and identify the smallest common multiple.​

3. What is the relationship between the LCM and HCF of 12, 15, and 21?

The Highest Common Factor (HCF) of 12, 15, and 21 is 3. The product of the LCM and HCF of two numbers equals the product of the numbers themselves. For multiple numbers, this relationship can be extended accordingly.

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