What is the LCM of 10, 15, 20 and 30?

To find the least common multiple (LCM) of the numbers 10, 15, 20, and 30, we can use a systematic approach that involves prime factorization. The LCM is the smallest number that is a multiple of each of the given numbers. Let's break it down step by step.

Step 1: Prime Factorization

First, we need to find the prime factorization of each number:

• 10: 2 × 5
• 15: 3 × 5
• 20: 2² × 5
• 30: 2 × 3 × 5

Step 2: Identify the Highest Powers

Next, we take the highest power of each prime number that appears in the factorizations:

• Prime 2: The highest power is 2² (from 20).
• Prime 3: The highest power is 3¹ (from 15 and 30).
• Prime 5: The highest power is 5¹ (from all numbers).

Step 3: Calculate the LCM

Now, we multiply these highest powers together to find the LCM:

LCM = 2² × 3¹ × 5¹

Calculating this gives:

• 2² = 4
• 3¹ = 3
• 5¹ = 5

Now, multiply these results:

4 × 3 = 12

12 × 5 = 60

The Result

Thus, the least common multiple of 10, 15, 20, and 30 is 60.

Understanding the Concept

Finding the LCM is particularly useful in various applications, such as solving problems involving fractions or scheduling events. By determining the LCM, you can find a common timeframe or denominator that allows for easier calculations and comparisons.

In summary, the LCM of 10, 15, 20, and 30 is 60, achieved through prime factorization and identifying the highest powers of the primes involved. This method ensures that we account for all multiples effectively.