Find the least square number which is divisible by 3, 5, 6, and 9.

Options

A. 900
B. 90
C. 8100
D. 81

Answer

To find the least square number that is divisible by 3, 5, 6, and 9, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
First, we need to find the LCM of the numbers 3, 5, 6, and 9.

- The prime factorization of each number is:
- 3 = \(3^1\)
- 5 = \(5^1\)
- 6 = \(2^1 \times 3^1\)
- 9 = \(3^2\)

To find the LCM, we take the highest power of each prime factor that appears in the factorizations:
- For 2: \(2^1\) (from 6)
- For 3: \(3^2\) (from 9)
- For 5: \(5^1\) (from 5)

Thus, the LCM is:
\[ LCM = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90 \]

### Step 2: Determine the Least Square Number
Next, we need to find the least square number that is divisible by 90. A square number must have even powers of all prime factors.

From the LCM, we have:
- \(2^1\) (needs to be \(2^2\) to be even)
- \(3^2\) (already even)
- \(5^1\) (needs to be \(5^2\) to be even)

To make the powers even, we multiply by:
- \(2^{2-1} = 2^1\)
- \(5^{2-1} = 5^1\)

Thus, we multiply the LCM by \(2^1 \times 5^1 = 2 \times 5 = 10\).

### Step 3: Calculate the Least Square Number
Now we calculate:
\[ \text{Least Square Number} = LCM \times (2^1 \times 5^1) = 90 \times 10 = 900 \]

### Conclusion
The least square number which is divisible by 3, 5, 6, and 9 is **900**.

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