What Is The Least Common Multiple Of 50 25

Finding the Least Common Multiple (LCM) of 50 and 25

This article will guide you through finding the least common multiple (LCM) of 50 and 25. Understanding LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cycles and timing. We'll explore different methods to calculate the LCM, making this concept clear and easy to understand.

What is the Least Common Multiple (LCM)?

The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more numbers. In simpler terms, it's the smallest number that both numbers 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.

Methods for Finding the LCM of 50 and 25

We can use several methods to determine the LCM of 50 and 25. Let's explore two common approaches:

1. Listing Multiples:

This method involves listing the multiples of each number until we find the smallest multiple common to both.

• Multiples of 50: 50, 100, 150, 200, 250...
• Multiples of 25: 25, 50, 75, 100, 125...

As you can see, the smallest multiple that appears in both lists is 50. Therefore, the LCM of 50 and 25 is 50.

2. Prime Factorization Method:

This method uses the prime factorization of each number to find the LCM. Prime factorization breaks a number down into its prime number components (numbers only divisible by 1 and themselves).

• Prime factorization of 50: 2 x 5 x 5 = 2 x 5²
• Prime factorization of 25: 5 x 5 = 5²

To find the LCM using prime factorization, we take the highest power of each prime factor present in the factorizations:

• The highest power of 2 is 2¹
• The highest power of 5 is 5²

Multiply these together: 2¹ x 5² = 2 x 25 = 50

Therefore, the LCM of 50 and 25, using the prime factorization method, is also 50.

Understanding the Relationship Between LCM and GCD

The greatest common divisor (GCD) and the least common multiple (LCM) are closely related. The product of the LCM and GCD of two numbers always equals the product of the two numbers. Let's verify this with 50 and 25:

• GCD of 50 and 25: 25 (25 is the largest number that divides both 50 and 25 evenly)
• LCM of 50 and 25: 50 (as calculated above)
• Product of the numbers: 50 x 25 = 1250
• Product of LCM and GCD: 50 x 25 = 1250

The relationship holds true! This provides a useful cross-check for your LCM calculations.

Conclusion:

The least common multiple of 50 and 25 is 50. Both the listing multiples and prime factorization methods confirm this result. Understanding the LCM is vital in various mathematical contexts, and applying these methods will help you confidently solve similar problems. Remember to consider the relationship between LCM and GCD for a more comprehensive understanding of number theory.