The smallest positive integer, when multiplied by 392, the product is a perfect square, is

Options

A. 6
B. 5
C. 3
D. 2

Answer

To find the smallest positive integer that, when multiplied by 392, results in a perfect square, we can follow these steps:

### Step 1: Prime Factorization of 392
First, we need to find the prime factorization of 392.

- Divide 392 by 2:
\[ 392 \div 2 = 196 \]
- Divide 196 by 2:
\[ 196 \div 2 = 98 \]
- Divide 98 by 2:
\[ 98 \div 2 = 49 \]
- Divide 49 by 7:
\[ 49 \div 7 = 7 \]
- Divide 7 by 7:
\[ 7 \div 7 = 1 \]

So, the prime factorization of 392 is:
\[ 392 = 2^3 \times 7^2 \]

### Step 2: Identify the Exponents
Next, we look at the exponents in the prime factorization:
- For \(2^3\), the exponent is 3 (which is odd).
- For \(7^2\), the exponent is 2 (which is even).

### Step 3: Make Exponents Even
To form a perfect square, all exponents in the prime factorization must be even.

- The exponent of 2 is 3 (odd), so we need to multiply by \(2^1\) to make it even (3 + 1 = 4).
- The exponent of 7 is already even, so we do not need to multiply by anything for 7.

### Step 4: Calculate the Required Integer
Thus, the smallest positive integer we need to multiply by 392 to make it a perfect square is:
\[ 2^1 = 2 \]

### Final Answer
The smallest positive integer that, when multiplied by 392, results in a perfect square is:
\[ \boxed{2} \]

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