LCM of 7 and 9

Introduction

When working with numbers, we often come across the need to find the Least Common Multiple (LCM). The LCM of two numbers is the smallest number that is evenly divisible by both. For this article, let's discover how to find the LCM of 7 and 9.

What is the LCM of 7 and 9?

The LCM is a very useful concept when dealing with fractions, common denominators, and solving problems involving multiple groups of items. The LCM of 7 and 9 is the smallest number that is a multiple of both 7 and 9.

Ways to Calculate the LCM

There are several ways to calculate the LCM of two numbers. Here are some methods we can use:

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

Prime Factorization Method

In the Prime Factorization Method, we break down each number into its prime factors.

1. Prime Factorization of 7: 7 is a prime number, so its prime factorization is simply 7.
2. Prime Factorization of 9: 9 is 3 × 3, so its prime factorization is 3 × 3.

Now, for the LCM, we take the highest power of each prime number involved.

• Highest power of 3: 3^2 (from 9)
• Highest power of 7: 7^1 (from 7)

We multiply these together to get the LCM:
3^2 × 7^1 = 9 × 7 = 63

So, the LCM of 7 and 9 is 63.

Division Method

In the Division Method, we repeatedly divide the numbers by their common prime factors until we can't divide anymore.

1. List down the numbers 7 and 9.

2. Begin with the smallest prime number, which is 2. Both numbers are not divisible by 2.

3. Move to the next prime number, which is 3. Only 9 is divisible by 3:

   • 7 (unchanged)
   • 9 divided by 3 gives 3

4. Again, 7 is not divisible by 3 but 3 is divisible by 3:

   • 7 (unchanged)
   • 3 divided by 3 gives 1

5. Now we have 7 and 1. We stop when one of the numbers reaches 1. The LCM is the product of all the divisors we used:

63 (from 3 × 3 × 7 = 63)

So, the LCM of 7 and 9 is 63.

Listing Method

In the Listing Method, we list the multiples of each number and find the smallest common one.

1. Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70…
2. Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72…

The smallest common multiple in both lists is 63.

So, the LCM of 7 and 9 is 63.

Formula for Calculating LCM

There is also a simple formula to calculate the LCM of two numbers using their Greatest Common Divisor (GCD):

LCM(a, b) = (a × b) / GCD(a, b)

For 7 and 9, since their GCD is 1:

LCM(7, 9) = (7 × 9) / 1 = 63

Conclusion

Finding the LCM of two numbers like 7 and 9 is straightforward using various methods. Whether we use prime factorization, division, or listing multiples, we arrive at the same answer, which is 63.

FAQs

What is the LCM of 7 and 9?

The LCM of 7 and 9 is 63.

Can the LCM be smaller than either of the numbers?

No, the LCM is always equal to or larger than the largest number.

Why do we use the LCM?

We use the LCM to solve problems involving fractions, find common denominators, and make calculations easier in various math problems.

Is the LCM the same as the GCD?

No, the LCM is the smallest multiple common to both numbers, while the GCD is the largest common factor.

What is the fastest method to find the LCM?

The fastest method often depends on the numbers involved. For small numbers, listing multiples can be quick, while for larger numbers, prime factorization or the formula method might be faster.