Find the smallest perfect square divisible by 3, 4, 5 and 6.

Answer

To find the smallest perfect square divisible by 3, 4, 5, and 6, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
First, we need to find the LCM of the numbers 3, 4, 5, and 6.

- **Prime factorization**:
- 3 = 3^1
- 4 = 2^2
- 5 = 5^1
- 6 = 2^1 × 3^1

Now, we take the highest power of each prime number:
- For 2: the highest power is 2^2 (from 4)
- For 3: the highest power is 3^1 (from 3 or 6)
- For 5: the highest power is 5^1 (from 5)

So, the LCM is:
\[ \text{LCM} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]

### Step 2: Factor the LCM
Now we have:
\[ 60 = 2^2 \times 3^1 \times 5^1 \]

### Step 3: Make it a Perfect Square
To make this a perfect square, we need to ensure that all the prime factors have even powers.

- The power of 2 is already even (2).
- The power of 3 is 1 (odd), so we need to increase it by 1 to make it 2.
- The power of 5 is 1 (odd), so we also need to increase it by 1 to make it 2.

Thus, we need to multiply the LCM by \(3^1\) and \(5^1\):
\[ \text{Required number} = 60 \times 3 \times 5 = 60 \times 15 = 900 \]

### Step 4: Verify if 900 is a Perfect Square
Now we check if 900 is a perfect square:
- The prime factorization of 900 is:
\[ 900 = 30^2 = (2^2 \times 3^2 \times 5^2) \]

Since all the prime factors have even powers, 900 is indeed a perfect square.

### Conclusion
Thus, the smallest perfect square divisible by 3, 4, 5, and 6 is **900**.

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