Let $N=2^{10}\cdot 3^5$. What is the smallest positive integer whose product with $N$ is a perfect square?

The smallest positive integer is 3.

Step-by-step explanation:

In order to detect if a number is a perfect square, you firstly have to write the number as a product of its prime factor. If the prime factors of the number appear in an even number of times, then it's a perfect square. For example, if we want to detect if 36 is a perfect square we have to write 36 as a product of its prime factors. Doing this, we have:

36 = 2 x 2 x 3 x 3 = 2² x 3². From here, we see that the prime factors of 36 – 2 and 3 – appear twice which is an even number of times. Hence, it's a perfect square.

Now let us go back to our number:

N = 2^10 x 3^5

Obviously, N is not a perfect square because one of it's prime factors, 3, appears 5 times which is an odd number of times. In order to make it a perfect square, we have to multiply it by a certain number of 3's such that it will appear in an even number of times. Since the least positive integer is required, then we have to multiply by the least number of 3 that will make 3^5 even. Obviously, one 3 is the least needed to make 3^5 appear even number of times, because 3^5 x 3 = 3^6.

Therefore we have

N x 3 = 2^10 x 3^5 x 3 = 2^10 x 3^6.