The least integer whose multiplication with 588 leads to a perfect square is

To find the least integer that makes 588 a perfect square when multiplied, we must examine its prime factorization. The prime factorization of 588 is 2!! 2!! 3!! 7!! 7, which can be expressed in exponential form as 2!!!! 3!!!! 7!!. For a number to be a perfect square, every prime factor in its factorization must have an even exponent. In the case of 588, the prime factors 2 and 7 already have even exponents (2 each), but the prime factor 3 has an exponent of 1. Therefore, to make all exponents even, we must multiply 588 by another 3. Multiplying 588 by 3 results in 1764, which is the square of 42 (2!! 3!! 7). Thus, the smallest integer required is 3.