What Is The Least Common Multiple Of 25 And 35

Finding the Least Common Multiple (LCM) of 25 and 35

This article will guide you through the process of calculating the least common multiple (LCM) of 25 and 35. Understanding LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cycles and periodic events. We'll explore different methods to find the LCM, ensuring you grasp the concept thoroughly.

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 contains all the numbers as factors. For example, the LCM of 2 and 3 is 6 because 6 is the smallest number divisible by both 2 and 3.

Method 1: Listing Multiples

This method is straightforward, especially for smaller numbers. We list the multiples of each number until we find the smallest multiple common to both.

• Multiples of 25: 25, 50, 75, 100, 125, 150, 175, ...
• Multiples of 35: 35, 70, 105, 140, 175, ...

Notice that 175 is the smallest number appearing in both lists. Therefore, the LCM of 25 and 35 is 175. This method works well for smaller numbers but becomes less efficient with larger numbers.

Method 2: Prime Factorization

This method is more efficient for larger numbers. We find the prime factorization of each number and then construct the LCM using the highest powers of each prime factor present.

• Prime factorization of 25: 5 x 5 = 5²
• Prime factorization of 35: 5 x 7

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

• The highest power of 5 is 5² = 25
• The highest power of 7 is 7¹ = 7

Therefore, the LCM of 25 and 35 is 25 x 7 = 175.

Method 3: Using the Formula (for two numbers)

For two numbers 'a' and 'b', the LCM can be calculated using the formula:

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

Where GCD is the Greatest Common Divisor.

First, we find the GCD of 25 and 35. The greatest common divisor of 25 and 35 is 5.

Then, applying the formula:

LCM(25, 35) = (25 * 35) / 5 = 875 / 5 = 175

Therefore, using this formula, we confirm that the LCM of 25 and 35 is 175.

Conclusion:

We've explored three different methods to determine the least common multiple of 25 and 35. Regardless of the method used, the LCM remains consistent at 175. Choosing the most efficient method depends on the numbers involved. For smaller numbers, listing multiples might suffice; however, prime factorization and the formula are more efficient for larger numbers. Understanding the LCM is fundamental to various mathematical concepts and problem-solving.