The product of two numbers X and Y is 3 less than the sum of the numbers. How many such integer pairs of X and Y are possible.

First we need to write the problem in mathematical language i.e. we need to write an equation.

Product of two numbers  sum of two numbers  and since the product is less by 3, we have

Write  in terms of 

Factor the left side.

Check whether  gives an integer solution.

 

 does not gives us an integer solution so we can assume that  which allows us to divide by

Now we know that in order to get an integer solution   must be divisible by   bearing in mind that  is an integer as well. The only thing remaining to do is to check whether those two numbers are divisible for some values of   We will start with values of  such that  or  because those are certainly integers.

 is not a solution. If we keep trying with the smaller and smaller numbers we will get fractions that get closer and closer to 1, but we will never get an integer.

Let us try greater numbers. We already know that  is not a solution so we will skip that.

Again, we see that  is not a solution and nor is any integer greater that 4.

We can conclude there are 4 such integer pair numbers and they are

 and