If you want to prove that formally, you can. Suppose you have a collection of numbers whose product is P. And suppose you have two numbers in the collection, a and b such that a < b. If you increment a, you change the product from P to P(a+1)/a. If you increment b, you change the product to P(b+1)/b. Since a < b, we know that (a+1)/a > (b+1)/b. (If you are stuck on that, think about examples: 3/2 is definitely greater than 101/100, right?) That means incrementing a will increase the factor by the greatest amount.
So, you need to find the minimum element and increment it. I would do that by sorting the vector and then incrementing the first element.
Now how about calculating the product? Since you are told that the product will be ≤ 262, you need to use a 64-bit integer. In C++, this is a long long. Calculate your product using that, and you're done.