The least common multiple of 18, 24, 60 will be:

Question

Accepted Answer

The least common multiple (LCM) of a set of numbers is the smallest positive integer that is divisible by each of the numbers in the set. To find the LCM of 18, 24, and 60, we can use the prime factorization method.

1. **Prime Factorization:**
   - 18 = 2 × 3²
   - 24 = 2³ × 3
   - 60 = 2² × 3 × 5

2. **Determine the LCM:**
   - For the LCM, take the highest power of each prime number that appears in the factorizations.
   - The prime numbers involved are 2, 3, and 5.
   - The highest power of 2 is 2³ (from 24).
   - The highest power of 3 is 3² (from 18).
   - The highest power of 5 is 5¹ (from 60).

3. **Calculate the LCM:**
   - LCM = 2³ × 3² × 5¹
   - LCM = 8 × 9 × 5
   - LCM = 72 × 5
   - LCM = 360

Thus, the least common multiple of 18, 24, and 60 is 360, which corresponds to Option 3.

**Explanation for Incorrect Options:**

- **Option 1 (120):** 
  - 120 is not divisible by 24 (since 120 ÷ 24 = 5, which is not an integer). Therefore, 120 cannot be the LCM of these numbers.

- **Option 2 (240):**
  - 240 is not divisible by 18 (since 240 ÷ 18 = 13.33, which is not an integer). Therefore, 240 cannot be the LCM of these numbers.

- **Option 4 (480):**
  - Although 480 is divisible by 18, 24, and 60, it is not the smallest number that satisfies this condition. Since 360 is smaller and also divisible by all three numbers, 480 cannot be the LCM.

In conclusion, Option 3 (360) is the correct answer because it is the smallest number that is divisible by 18, 24, and 60, fulfilling the definition of the least common multiple.