LCM of 80,85 and 90

Introduction

The Least Common Multiple (LCM) of a set of numbers is the smallest number that is a multiple of all the numbers in the set. Understanding how to find the LCM of numbers can be very helpful in solving a variety of math problems, from fractions to word problems. Today, let's explore how to find the LCM of 80, 85, and 90 using various methods.

What Is the LCM of 80, 85, and 90?

The LCM of 80, 85, and 90 is the smallest number that all three numbers divide into without leaving any remainder. Knowing how to find the LCM is useful in many situations, such as adding or subtracting fractions and solving problems that involve repeating events.

Ways to Calculate the LCM

There are different methods to calculate the LCM. Here are three common ones:

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

Prime Factorization Method

One way to find the LCM is by using the prime factorization method. Here's how you can do it:

1. Find the prime factorization of each number.
2. Write down all the prime factors, taking the highest power of each prime that appears in the factorizations.
3. Multiply these highest powers together to get the LCM.

Prime factorization for each number:

• 80 = 2^4 * 5
• 85 = 5 * 17
• 90 = 2 * 3^2 * 5

Now, take the highest powers:

• 2^4
• 3^2
• 5
• 17

Multiply them together:
LCM = 2^4 * 3^2 * 5 * 17 = 2 * 2 * 2 * 2 * 3 * 3 * 5 * 17 = 12240

Division Method

The division method involves dividing the numbers simultaneously by their common prime factors until only 1's are left. Here are the steps:

1. Write down the numbers 80, 85, and 90 in a row.
2. Divide them by the smallest prime number that can divide at least one of the numbers.
3. Write the quotient below each number and repeat the process with the new set of numbers.
4. Continue dividing until you reach 1's for all numbers.

For 80, 85, and 90:

2 | 80, 85, 90
2 | 40, 85, 45
2 | 20, 85, 45
2 | 10, 85, 45
5 | 5, 85, 45
5 | 1, 17, 9
3 | 1, 17, 3
17 | 1, 17, 1
1 | 1, 1, 1

Multiply all the divisors: 2 * 2 * 2 * 2 * 5 * 17 * 3 = 12240

Listing Method

The listing method involves listing the multiples of each number until we find the smallest multiple common to all of them. Here are the steps:

1. Write down a list of multiples for each number.
2. Identify the smallest multiple that appears in all lists.

Multiples of 80: 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200, 1280, 1360, 1440, 1520, 1600, …
Multiples of 85: 85, 170, 255, 340, 425, 510, 595, 680, 765, 850, 935, 1020, 1105, 1190, 1275, 1360, 1445, 1530, 1615, …
Multiples of 90: 90, 180, 270, 360, 450, 540, 630, 720, 810, 900, 990, 1080, 1170, 1260, 1350, 1440, 1530, 1620, 1710, …

The smallest common multiple is 12240.

Formula for Calculating LCM

The formula for calculating the LCM of two numbers (a and b) using the greatest common divisor (GCD) is:

LCM(a, b) = abs(a * b) / GCD(a, b)

Conclusion

Finding the LCM of 80, 85, and 90 can be done in several ways, including the prime factorization method, the division method, and the listing method. Each method offers a unique approach to solving the problem, but all lead to the same result, which is 12240.

FAQs

1. What is the LCM of 80, 85, and 90?

   • The LCM of 80, 85, and 90 is 12240.

2. Why is finding the LCM important?

   • Finding the LCM is important in many areas of math, such as solving problems involving fractions and predicting repeating events.

3. Is there a formula to easily calculate the LCM?

• Yes, the formula is LCM(a, b) = abs(a * b) / GCD(a, b).

4. Can you find the LCM using only pen and paper?
   • Yes, methods like listing multiples and prime factorization can be done using only pen and paper.