The product of two natural numbers is 1000. None of them is divisible by 10. Find their sum.

Two numbers make up the factors of 1000. In order that neither of the numbers be divisible by ten, each must contain powers of only one of the prime factors of 10, namely, 2 OR 5.

So let's see what we can cook up. The obvious answer is

1000 = 10 * 10 * 10

1000 = 2*5 * 2*5 * 2*5

1000 = 2^3 * 5^3

1000 = 8 * 125 Are there any others?

I don't think so. The answer then comes to 8 + 125 = 133

You might argue that - 8 * - 125 would also work. But the question says natural numbers not integers.