The least perfect square number which is divisible by each of 21,36 and 66 is

Options

A. 213444
B. 214344
C. 214434
D. 231444

Answer

To find the least perfect square number that is divisible by each of the numbers 21, 36, and 66, we will follow these steps:

### Step 1: Find the prime factorization of each number.
- **21**: The prime factorization of 21 is \(3^1 \times 7^1\).
- **36**: The prime factorization of 36 is \(2^2 \times 3^2\).
- **66**: The prime factorization of 66 is \(2^1 \times 3^1 \times 11^1\).

### Step 2: Determine the LCM of the numbers.
To find the LCM, we take the highest power of each prime factor that appears in the factorizations:
- For \(2\): The highest power is \(2^2\) (from 36).
- For \(3\): The highest power is \(3^2\) (from 36).
- For \(7\): The highest power is \(7^1\) (from 21).
- For \(11\): The highest power is \(11^1\) (from 66).

Thus, the LCM is:
\[ LCM = 2^2 \times 3^2 \times 7^1 \times 11^1 \]

### Step 3: Calculate the LCM.
Now we calculate the LCM:
\[ LCM = 4 \times 9 \times 7 \times 11 \]
Calculating step by step:
1. \(4 \times 9 = 36\)
2. \(36 \times 7 = 252\)
3. \(252 \times 11 = 2772\)

So, the LCM of 21, 36, and 66 is \(2772\).

### Step 4: Make the LCM a perfect square.
A perfect square requires all prime factors to have even powers. The prime factorization of \(2772\) is:
\[ 2772 = 2^2 \times 3^2 \times 7^1 \times 11^1 \]
To make this a perfect square, we need to adjust the powers of \(7\) and \(11\):
- \(7^1\) needs one more \(7\) to become \(7^2\).
- \(11^1\) needs one more \(11\) to become \(11^2\).

### Step 5: Multiply by the necessary factors.
To make \(2772\) a perfect square, we multiply it by \(7^1\) and \(11^1\):
\[ \text{Required number} = 2772 \times 7 \times 11 \]

### Step 6: Calculate the least perfect square.
Now we calculate:
\[ 2772 \times 7 = 19404 \]
\[ 19404 \times 11 = 213444 \]

Thus, the least perfect square number which is divisible by each of 21, 36, and 66 is \(213444\).

### Final Answer:
The least perfect square number which is divisible by each of 21, 36, and 66 is **213444**.

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