The smallest positive integer which when multiplied by 392 , gives a perfect square is (a) 2 (b) 3 (c) 5 (d) 7

To find the smallest positive integer that, when multiplied by 392, yields a perfect square, we first need to factorize 392 to understand its prime factorization.

Prime Factorization of 392

\[
392 = 2^3 \times 7^2
\]

For a number to be a perfect square, all the exponents in its prime factorization must be even. In the prime factorization of 392, the exponent of 2 is 3 (which is odd), and the exponent of 7 is 2 (which is even).

Finding the Smallest Positive Integer

To make the number a perfect square, we need to multiply it by a number that will make all the exponents even. Since the exponent of 2 is odd, we need to multiply 392 by another 2 to make the exponent of 2 become \(3 + 1 = 4\), which is even.

\[
392 \times 2 = 2^4 \times 7^2
\]

Now, the prime factorization \(2^4 \times 7^2\) indicates a perfect square because both exponents are even.

Conclusion

The smallest positive integer that, when multiplied by 392, yields a perfect square is \(2\).

Answer: (a) 2.