What Is The Least Common Multiple Of 120 And 80

Finding the Least Common Multiple (LCM) of 120 and 80

Finding the least common multiple (LCM) is a fundamental concept in mathematics, particularly useful in various fields like scheduling and fractions. This article will guide you through calculating the LCM of 120 and 80 using two common methods: prime factorization and the greatest common divisor (GCD) method. Understanding these methods will equip you with the skills to find the LCM of any two numbers.

What is the Least Common Multiple (LCM)?

The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by all the integers. In simpler terms, it's the smallest number that both numbers can divide into evenly. For example, the LCM of 2 and 3 is 6, because 6 is the smallest number divisible by both 2 and 3.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors. The prime factorization is the expression of a number as a product of its prime numbers. Let's find the prime factorization of 120 and 80:

• 120: 2 x 2 x 2 x 3 x 5 = 2³ x 3 x 5
• 80: 2 x 2 x 2 x 2 x 5 = 2⁴ x 5

Now, to find the LCM, we take the highest power of each prime factor present in either factorization and multiply them together:

LCM(120, 80) = 2⁴ x 3 x 5 = 16 x 3 x 5 = 240

Therefore, the least common multiple of 120 and 80 is 240.

Method 2: Using the Greatest Common Divisor (GCD)

This method leverages the relationship between the LCM and GCD (Greatest Common Divisor) of two numbers. The formula is:

LCM(a, b) = (|a x b|) / GCD(a, b)

First, we need to find the GCD of 120 and 80. We can use the Euclidean algorithm for this:

1. Divide the larger number (120) by the smaller number (80): 120 ÷ 80 = 1 with a remainder of 40
2. Replace the larger number with the smaller number (80) and the smaller number with the remainder (40): 80 ÷ 40 = 2 with a remainder of 0

Since the remainder is 0, the GCD is the last non-zero remainder, which is 40.

Now, we can use the formula:

LCM(120, 80) = (120 x 80) / 40 = 9600 / 40 = 240

Again, the least common multiple of 120 and 80 is 240.

Conclusion

Both methods demonstrate that the least common multiple of 120 and 80 is 240. Choosing the method depends on your preference and the complexity of the numbers involved. The prime factorization method is often easier to visualize, while the GCD method can be more efficient for larger numbers. Understanding both approaches provides a strong foundation for tackling various mathematical problems involving LCM. Remember to practice these methods to improve your skills in finding the LCM of different number pairs.