By what number must $192$ be multiplied, so that the product may be a perfect square?

Let's find the prime factorization of $192$:
$$192=2\times 96=2^2\times 48=2^3\times 24=2^4\times 12=2^5\times 6=2^6\times 3$$

Grouping the prime factors into pairs:

$$192=(2\times 2)\times (2\times 2)\times (2\times 2)\times 3=2^6\times 3^1$$

To make it a perfect square, every prime factor must have an even exponent. Therefore, we need to multiply by $3$ to complete the pair for $3$.