Least Common Multiple Of 15 And 45

Finding the Least Common Multiple (LCM) of 15 and 45

This article will guide you through the process of calculating the least common multiple (LCM) of 15 and 45. Understanding LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cyclical events. We'll explore different methods to find the LCM, ensuring you grasp the concept thoroughly. This will cover prime factorization, the listing method, and using the greatest common divisor (GCD).

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 15 and 45 divide into evenly. Knowing the LCM is helpful when dealing with fractions, finding common denominators, and solving problems involving repeating cycles or patterns.

Method 1: Prime Factorization

This method is efficient for finding the LCM of larger numbers. It involves breaking down each number into its prime factors.

1. Find the prime factorization of each number:

   • 15 = 3 x 5
   • 45 = 3 x 3 x 5 = 3² x 5

2. Identify the highest power of each prime factor:

   • The prime factors are 3 and 5.
   • The highest power of 3 is 3² = 9
   • The highest power of 5 is 5¹ = 5

3. Multiply the highest powers together:

   • LCM(15, 45) = 3² x 5 = 9 x 5 = 45

Therefore, the least common multiple of 15 and 45 is $\boxed{45}$.

Method 2: Listing Multiples

This method is straightforward for smaller numbers. List the multiples of each number until you find the smallest common multiple.

• Multiples of 15: 15, 30, 45, 60, 75, ...
• Multiples of 45: 45, 90, 135, ...

The smallest number that appears in both lists is 45. Therefore, the LCM(15, 45) = 45.

Method 3: Using the Greatest Common Divisor (GCD)

There's a relationship between the LCM and the greatest common divisor (GCD) of two numbers. The formula is:

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

1. Find the GCD of 15 and 45:

   • The factors of 15 are 1, 3, 5, 15.
   • The factors of 45 are 1, 3, 5, 9, 15, 45.
   • The greatest common factor is 15. Therefore, GCD(15, 45) = 15.

2. Apply the formula:

   • LCM(15, 45) = (15 x 45) / 15 = 45

Again, the LCM of 15 and 45 is $\boxed{45}$.

Conclusion:

We've demonstrated three different methods to find the least common multiple of 15 and 45, all resulting in the answer 45. Choosing the best method depends on the numbers involved. Prime factorization is generally more efficient for larger numbers, while the listing method is simpler for smaller numbers. Understanding the relationship between LCM and GCD provides another powerful tool for calculation. Mastering these techniques will enhance your problem-solving skills in various mathematical contexts.