The least perfect square which is divisible by 3, 4, 5, 6, 8 is :

Question

Accepted Answer

To determine the least perfect square divisible by 3, 4, 5, 6, and 8, we need to follow these steps:

1. **Find the Least Common Multiple (LCM):**
   - First, find the LCM of the numbers 3, 4, 5, 6, and 8.
   - Prime factorization:
     - 3 = 3
     - 4 = 2²
     - 5 = 5
     - 6 = 2 × 3
     - 8 = 2³
   - The LCM is found by taking the highest power of each prime number present in the factorizations:
     - LCM = 2³ × 3 × 5 = 120

2. **Determine the Least Perfect Square:**
   - A perfect square must have even powers of all prime factors.
   - The LCM 120 = 2³ × 3 × 5. To make it a perfect square, we need even powers:
     - For 2³, we need one more 2 to make it 2⁴.
     - For 3, we need one more 3 to make it 3².
     - For 5, we need one more 5 to make it 5².
   - Therefore, the smallest perfect square divisible by 120 is:
     - (2⁴ × 3² × 5²) = 16 × 9 × 25 = 3600

3. **Explanation of Options:**
   - **Option 1: 4900**
     - 4900 = 70² = (2 × 5 × 7)²
     - 4900 is not divisible by 3, so it cannot be the correct answer.
   - **Option 2: 1600**
     - 1600 = 40² = (2⁴ × 5)²
     - 1600 is not divisible by 3, so it cannot be the correct answer.
   - **Option 3: 2500**
     - 2500 = 50² = (2 × 5²)²
     - 2500 is not divisible by 3, so it cannot be the correct answer.
   - **Option 4: 3600**
     - 3600 = 60² = (2² × 3 × 5)²
     - 3600 is divisible by 3, 4, 5, 6, and 8, making it the correct answer.

Thus, Option 4 (3600) is the least perfect square divisible by 3, 4, 5, 6, and 8, as it satisfies all the conditions required.