The question asks for the least integer that, when multiplied by 588, will result in a perfect square. We can approach this problem by finding the prime factors of 588 and examining each option.

Option 1: Multiplying 2 by 588 gives us 1176, which is not a perfect square.

Option 2: Multiplying 3 by 588 gives us 1764, which is a perfect square (42^2). This matches the correct answer provided.

Option 3: Multiplying 4 by 588 gives us 2352, which is not a perfect square.

Option 4: Multiplying 7 by 588 gives us 4116, which is not a perfect square.

Therefore, the correct answer is option 2, where the least integer that, when multiplied by 588, leads to a perfect square, is 3.