Find the smallest number by which `396` must be multiplied so that the product becomes a perfect square.

Options

A. `13`
B. `11`
C. `12`
D. `15`

Answer

To find the smallest number by which 396 must be multiplied so that the product becomes a perfect square, we can follow these steps:

### Step 1: Prime Factorization of 396
First, we need to find the prime factorization of 396.

- Divide 396 by 2 (the smallest prime number):
- \( 396 \div 2 = 198 \)
- Divide 198 by 2 again:
- \( 198 \div 2 = 99 \)
- Now, divide 99 by 3 (the next smallest prime number):
- \( 99 \div 3 = 33 \)
- Divide 33 by 3 again:
- \( 33 \div 3 = 11 \)
- Finally, divide 11 by 11:
- \( 11 \div 11 = 1 \)

Thus, the prime factorization of 396 is:
\[ 396 = 2^2 \times 3^2 \times 11^1 \]

### Step 2: Identify the Exponents of the Prime Factors
Next, we look at the exponents of the prime factors:
- For \( 2^2 \), the exponent is 2 (which is even).
- For \( 3^2 \), the exponent is 2 (which is even).
- For \( 11^1 \), the exponent is 1 (which is odd).

### Step 3: Determine Which Factors Need to be Paired
For a number to be a perfect square, all the exponents in its prime factorization must be even. Here, the exponent of 11 is odd.

### Step 4: Find the Smallest Number to Multiply
To make the exponent of 11 even, we need to multiply by 11 (to make it \( 11^2 \)):
- Thus, the smallest number we need to multiply 396 by is \( 11 \).

### Step 5: Verify the Result
Now, let's verify:
- If we multiply \( 396 \) by \( 11 \):
\[ 396 \times 11 = 4356 \]
- Now, let's check the prime factorization of \( 4356 \):
\[ 4356 = 2^2 \times 3^2 \times 11^2 \]
- All the exponents are even, confirming that \( 4356 \) is indeed a perfect square.

### Conclusion
The smallest number by which 396 must be multiplied to become a perfect square is \( 11 \).

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