Find the smallest perfect square number which is exactly divisible by 4, 5, 6, 15 and 18 :

Question

Accepted Answer

To determine the smallest perfect square number that is exactly divisible by 4, 5, 6, 15, and 18, we need to follow these steps:

1. **Find the Least Common Multiple (LCM):**
   - First, we find the prime factorization of each number:
     - 4 = 2²
     - 5 = 5¹
     - 6 = 2 × 3
     - 15 = 3 × 5
     - 18 = 2 × 3²

- The LCM is found by taking the highest power of each prime number that appears in these factorizations:
     - Highest power of 2: 2²
     - Highest power of 3: 3²
     - Highest power of 5: 5¹

- Therefore, the LCM = 2² × 3² × 5 = 180.

2. **Convert the LCM to a Perfect Square:**
   - A perfect square must have even powers of all prime factors. The LCM of 180 is not a perfect square because:
     - 2² is already even.
     - 3² is already even.
     - 5¹ is odd.

- To make it a perfect square, we need to multiply by 5 to make the power of 5 even:
     - New number = 180 × 5 = 900.

3. **Verify that 900 is a Perfect Square:**
   - The prime factorization of 900 is:
     - 900 = 2² × 3² × 5²
   - All prime factors have even powers, confirming that 900 is a perfect square.

4. **Check Divisibility:**
   - 900 is divisible by 4, 5, 6, 15, and 18 because it is a multiple of their LCM, 180.

Thus, the smallest perfect square number that is exactly divisible by 4, 5, 6, 15, and 18 is indeed 900, which corresponds to Option 3.

**Explanation for Incorrect Options:**

- **Option 1 (800):** 
  - The prime factorization of 800 is 2⁵ × 5². It is not a perfect square because the power of 2 is odd.

- **Option 2 (225):**
  - The prime factorization of 225 is 3² × 5². While it is a perfect square, it is not divisible by 4 or 18.

- **Option 4 (361):**
  - The prime factorization of 361 is 19². It is a perfect square but not divisible by any of the given numbers except 1.

Therefore, Option 3 (900) is the correct answer as it satisfies all the conditions of being a perfect square and divisible by the given numbers.