LCM of 7,8,14 and 21

Certainly, let’s dive into the topic of finding the Least Common Multiple (LCM) of 7, 8, 14, and 21.

Introduction

When we deal with numbers in math, sometimes we need to find a common multiple that is the smallest one for all the numbers given. This is called the Least Common Multiple (LCM). For instance, in this article, we will look at different methods to find the LCM of the numbers 7, 8, 14, and 21.

What is LCM?

The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. For example, when looking at 7, 8, 14, and 21, we want to find the smallest number that all these numbers can divide into without leaving a remainder.

Ways to Calculate LCM

There are several ways to find the LCM of numbers. Here are some common methods:

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

Prime Factorization Method

The Prime Factorization Method involves breaking down each of the numbers into their prime factors. Here’s how we can do it for 7, 8, 14, and 21:

1. Prime factors of 7: 7
2. Prime factors of 8: 2 * 2 * 2
3. Prime factors of 14: 2 * 7
4. Prime factors of 21: 3 * 7

Now, we take the highest powers of all the prime numbers:

• 2^3 (from 8)
• 3^1 (from 21)
• 7^1 (common in 7, 14, and 21)

Multiply these together: 2^3 * 3^1 * 7 = 8 * 3 * 7 = 168

So, the LCM of 7, 8, 14, and 21 is 168.

Division Method

The Division Method involves dividing the numbers by common prime factors until only 1s are left. Here’s the process for 7, 8, 14, and 21:

1. Write the numbers in a line:
   7, 8, 14, 21

2. Divide by the smallest prime number that can divide at least one number in the list, continue the process:
   2 | 7, 8, 14, 21
   | 7, 4, 7, 21
   2 | 7, 4, 7, 21
   | 7, 2, 7, 21
   2 | 7, 2, 7, 21
   | 7, 1, 7, 21
   3 | 7, 1, 7, 21
   | 7, 1, 7, 7
   7 | 7, 1, 7, 7
   | 1, 1, 1, 1

3. Multiply the divisors used: 2 * 2 * 2 * 3 * 7 = 168

So, the LCM of 7, 8, 14, and 21 is 168.

Listing Method

The Listing Method involves listing the multiples of each number until a common multiple is found. Here’s how to do it for 7, 8, 14, and 21:

1. List the multiples of each number:

   • Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, …
   • Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, …
   • Multiples of 14: 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, …
   • Multiples of 21: 21, 42, 63, 84, 105, 126, 147, 168, …

2. The smallest common multiple for all numbers is 168.

So, the LCM of 7, 8, 14, and 21 is 168.

LCM Formula

There is also a formula for finding the LCM using prime factors:

For numbers a, b, c, and d:
LCM(a, b, c, d) = (Highest power of all prime factors)

Using prime factors, we get:
LCM(7, 8, 14, 21) = 2^3 * 3 * 7 = 168

Conclusion

Finding the Least Common Multiple (LCM) helps us work with multiple numbers more easily, like when we need to add fractions or synchronize repeating events. For the numbers 7, 8, 14, and 21, we found that the LCM is 168 using prime factorization, division, and listing methods.

FAQs

Q: What is the Difference between LCM and GCD?
A: LCM is the least common multiple, while GCD (Greatest Common Divisor) is the largest number that can divide two or more numbers without a remainder.

Q: Can LCM be Smaller than The Largest Number?
A: No, the LCM of given numbers will always be equal to or larger than the largest number.

Q: Can We Use LCM for More Than Two Numbers?
A: Yes, we can find the LCM for any number of given numbers.

Q: Why Is The LCM Important?
A: The LCM is useful in solving problems involving adding, subtracting, or comparing fractions and in scheduling events to happen simultaneously.

Q: How Can We Check If Our LCM Calculation is Correct?
A: Multiply the LCM by each of the original numbers. Each product should be divisible evenly by the other numbers.