lcm of 15 and 10

2025

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30-03-2025

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Finding the least common multiple (LCM) is a fundamental concept in mathematics, particularly useful in working with fractions and solving problems involving cycles or repetitions. This article will guide you through calculating the LCM of 15 and 10 using two common methods: listing multiples and using prime factorization.

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 they have in common.

Step 1: List the multiples of 15:

15, 30, 45, 60, 75, 90, 105, 120...

Step 2: List the multiples of 10:

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120...

Step 3: Identify the smallest common multiple:

Notice that both lists share the number 30 and 60. However, 30 is the smallest number appearing in both lists.

Therefore, the LCM of 15 and 10 is $\boxed{30}$.

Method 2: Prime Factorization

This method is more efficient for larger numbers. It involves breaking down each number into its prime factors.

Step 1: Find the prime factorization of 15:

15 = 3 × 5

Step 2: Find the prime factorization of 10:

10 = 2 × 5

Step 3: Identify the highest power of each prime factor:

The prime factors involved are 2, 3, and 5. The highest power of 2 is 2¹, the highest power of 3 is 3¹, and the highest power of 5 is 5¹.

Step 4: Multiply the highest powers together:

LCM(15, 10) = 2¹ × 3¹ × 5¹ = 2 × 3 × 5 = 30

Therefore, the LCM of 15 and 10 is again $\boxed{30}$.

Which Method is Best?

The listing multiples method is simpler for small numbers. However, the prime factorization method is significantly more efficient and less error-prone for larger numbers where listing all multiples becomes impractical. Understanding both methods provides valuable flexibility in solving LCM problems.

Real-World Applications of LCM

Finding the LCM has practical applications in various fields:

• Scheduling: Determining when events with different repeating cycles will occur simultaneously. For example, if two buses arrive at a stop every 15 and 10 minutes respectively, the LCM helps find when they'll arrive together.

• Fractions: Finding the least common denominator when adding or subtracting fractions.

• Construction and Engineering: Calculating the lengths needed for materials or aligning patterns.

This exploration of the LCM of 15 and 10 provides a solid foundation for understanding this important mathematical concept and its practical applications. Remember to choose the method best suited to the numbers involved.