The Least Common Multiple Of 14 And 21

LCM of 14 and 21 How to Find LCM of 14, 21? from www.cuemath.com

Introduction

The concept of finding the least common multiple of two numbers is an important one in mathematics. The least common multiple (LCM) is the smallest number that is a multiple of both of the given numbers. In this article, we will explore how to find the LCM of 14 and 21.

What is a Multiple?

Before we can talk about the LCM, it's important to understand what a multiple is. A multiple of a number is the result of multiplying that number by another whole number. For example, the multiples of 3 are 3, 6, 9, 12, and so on.

Prime Factorization

To find the LCM of 14 and 21, we need to factor both numbers into their prime factors. Prime factors are the prime numbers that can be multiplied together to get the original number. For example, the prime factors of 12 are 2, 2, and 3.

Prime Factors of 14

To find the prime factors of 14, we can start by dividing it by the smallest prime number, which is 2. We get: 14 ÷ 2 = 7 Since 7 is a prime number, we can stop here. Therefore, the prime factors of 14 are 2 and 7.

Prime Factors of 21

To find the prime factors of 21, we can again start by dividing it by 2. However, 21 is not divisible by 2, so we move on to the next prime number, which is 3. We get: 21 ÷ 3 = 7 Again, since 7 is a prime number, we can stop here. Therefore, the prime factors of 21 are 3 and 7.

LCM

Now that we have the prime factors of both 14 and 21, we can find their LCM. To do this, we need to find the highest power of each prime number that appears in either factorization, and multiply them together.

Highest Power of 2

The prime factorization of 14 is 2 × 7, and the prime factorization of 21 is 3 × 7. The highest power of 2 that appears in either factorization is 2^1 (since 2 appears once in the factorization of 14 and not at all in the factorization of 21).

Highest Power of 3

The highest power of 3 that appears in either factorization is 3^1 (since 3 appears once in the factorization of 21 and not at all in the factorization of 14).

Highest Power of 7

The highest power of 7 that appears in either factorization is 7^1 (since 7 appears once in the factorization of both 14 and 21).

Multiplying the Highest Powers

Now that we have found the highest power of each prime number that appears in either factorization, we can multiply them together to get the LCM: 2^1 × 3^1 × 7^1 = 42 Therefore, the LCM of 14 and 21 is 42.

Conclusion

In conclusion, finding the LCM of two numbers requires finding their prime factorizations and then multiplying the highest powers of each prime number. For 14 and 21, the LCM is 42. This concept is important in many areas of mathematics and can be useful in solving various problems.