Number Theory problem #1417

Assuming that we have integers such that $ab = 1000$ , then since $(a-b)^2 + 4ab = (a+b)^2$ , so $(a-b)^2 = (a+b)^2 - 4000$ . It follows that finding the smallest possible sum $(a+b)$ is equivalent to finding the smallest possible difference $|a-b|$ of these integers.

We have $1000 = 2^3 \times 5^3$ , and $\sqrt{1000} \approx 32$ . By listing out the divisors, we can see that the closest divisors of 1000 to 32 are 25 and 40. Thus, the 2 integers with product 1000 and smallest possible difference will be 25 and 40. Hence the sum is $25 + 40 = 65$ .

If we are interested in $xy = 1000 = 2^{3}5^{3},$ then we have the following divisor pairs in positive integers:

$(x,y) = (1,1000); (2,500); (4,250); (5,200); (8,125); (10,100); (20,50); (25,40)$

The minimum sum of $x+y$ occurs at $(x,y) = (25,40)$ ), or $\boxed{65}.$