Write a pair of integers whose product is -12 and their lies 7 integers between them.
Answers
Answer:
Originally, we are asked about the number of integers between y and x exclusive, but nothing changes if we move everything over by some fixed amount, so let’s subtract y (the smaller value) from everything. Now the question is about the integers between 0 and x - y. These are 1, 2, 3, …, x - y - 1. How many is that? Well, these numbers count themselves! There are clearly x - y - 1 many of them.
You can also think about it visually. Picture a number line, with the integers as dots, and edges connecting adjacent integers. x - y is of course the number of edges from y to x. The larger endpoint on each of these edges is either an integer between y and x exclusive, or x itself (in the case of the last edge). Ignoring that last edge, there are x - y - 1 many edges, and they are in correspondence with the integers between x and y; that is to say, there are x - y - 1 many integers between y and x.
(This visualization is also commonly described in terms of fenceposts and slats of fence between them; hence terminology like “fencepost error” for getting this calculation off by one.)