Find the LCM of the following numbers: 24, 36, 45, and 60.

To find the least common multiple (LCM) of the numbers 24, 36, 45, and 60, we can use the prime factorization method. This approach involves breaking down each number into its prime factors and then using these factors to determine the LCM. Let’s walk through the steps together.

Step 1: Prime Factorization of Each Number

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

• 24: The prime factors are 2 × 2 × 2 × 3, which can be written as 2³ × 3¹.
• 36: The prime factors are 2 × 2 × 3 × 3, or 2² × 3².
• 45: The prime factors are 3 × 3 × 5, which is 3² × 5¹.
• 60: The prime factors are 2 × 2 × 3 × 5, or 2² × 3¹ × 5¹.

Step 2: Identify the Highest Powers of Each Prime Factor

Next, we need to identify the highest power of each prime factor that appears in any of the factorizations:

• For the prime factor 2, the highest power is 2³ (from 24).
• For the prime factor 3, the highest power is 3² (from both 36 and 45).
• For the prime factor 5, the highest power is 5¹ (from both 45 and 60).

Step 3: Calculate the LCM

Now, we can calculate the LCM by multiplying these highest powers together:

LCM = 2³ × 3² × 5¹

Calculating this step-by-step:

• 2³ = 8
• 3² = 9
• 5¹ = 5

Now, multiply these results together:

8 × 9 = 72

72 × 5 = 360

The Final Result

Thus, the least common multiple of 24, 36, 45, and 60 is 360.

This method is effective because it ensures that the LCM is the smallest number that all original numbers can divide into without leaving a remainder. If you have any further questions about LCM or related topics, feel free to ask!