Find the smallest number by which `252` must be multiplied to get a perfect square. Also, find the square root of the perfect square so obtained.

Answer

To solve the problem of finding the smallest number by which 252 must be multiplied to get a perfect square, and also to find the square root of the perfect square obtained, we can follow these steps:

### Step 1: Factor the number 252
First, we need to find the prime factorization of 252.

- Divide 252 by 2:
\( 252 \div 2 = 126 \)
- Divide 126 by 2:
\( 126 \div 2 = 63 \)
- Divide 63 by 3:
\( 63 \div 3 = 21 \)
- Divide 21 by 3:
\( 21 \div 3 = 7 \)
- Finally, 7 is a prime number.

So, the prime factorization of 252 is:
\[ 252 = 2^2 \times 3^2 \times 7^1 \]

### Step 2: Identify the factors for forming pairs
In order to form a perfect square, all prime factors must have even powers.

- From the factorization \( 2^2 \times 3^2 \times 7^1 \):
- The powers of 2 and 3 are already even (2 and 2).
- The power of 7 is odd (1).

### Step 3: Determine the smallest number to multiply
To make the power of 7 even, we need to multiply by one more 7. Thus, we need to multiply 252 by 7 to make it a perfect square.

So, the smallest number by which 252 must be multiplied is:
\[ 7 \]

### Step 4: Calculate the perfect square
Now, we multiply 252 by 7:
\[ 252 \times 7 = 1764 \]

### Step 5: Find the square root of the perfect square
Next, we need to find the square root of 1764. We can use the prime factorization we found earlier:

- The prime factorization of 1764 is:
\[ 1764 = 2^2 \times 3^2 \times 7^2 \]

To find the square root, we take half of the powers of the prime factors:
\[ \sqrt{1764} = 2^{2/2} \times 3^{2/2} \times 7^{2/2} = 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 \]

Calculating this gives:
\[ 2 \times 3 = 6 \]
\[ 6 \times 7 = 42 \]

Thus, the square root of the perfect square 1764 is:
\[ 42 \]

### Final Answer
- The smallest number by which 252 must be multiplied to get a perfect square is: **7**
- The square root of the perfect square obtained (1764) is: **42**

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