Given positive integer n, consider the set of numbers {n²+1, n²+2, ... (n+1)²}. If we pick two numbers x and y out of that set, how many different values can the product xy take?

The number of terms of the set is:

(n+1)^2 - (n^2 + 1) + 1 = n^2 + 2n + 1 - n^2 - 1 = 2n + 1

So we have (2n+1) sequential positive integers.

Since each term multiplied by the others always give a different product, the values so obtained are the combination of (2n+1) taken two by two:

X = (2n+1)*2n/2 = n(2n+1)

P.S. FK, the rating "1" is not mine. When I open this problem, it was already there. It seems to me that somebody is visiting all non-rated problems and giving a rate of "1". It happened with 2 or 3 problems that I posted.

 

Posted by pcbouhid on 2005-08-16 20:28:52