I pick two whole numbers x and y between 1 and 10 inclusive (not necessarily distinct). My friend picks two numbers x - 4 and 2y + 1. If the product of my friend's numbers is one greater than the product of my numbers, then what is the product of my numbers?
Let's see what would happen if we write the problem in mathematical expression.
\((2y + 1)(x - 4) = xy + 1\\ 2xy + x - 8y - 4 = xy + 1\\ xy + x - 8y - 5 = 0\)
This is still very hard to manipulate, but let us do it like this:
\(xy + x - 8y - 8 + 3 = 0\\ xy + x - 8y - 8 = -3\\ x(y + 1) - 8(y + 1) = -3\\ (x - 8)(y + 1) = -3 \)
Since we get a negative result, we know that x is smaller than 8. We can write it as \((8 - x)(y + 1) = 3\) by multiplying both sides by -1.
Now, since 8 - x and y + 1 are integers, there are only two possibilities: (i) the left-hand side is \(1 \times 3\), (ii) the left-hand side is \(3\times 1\). Which one is 1 and which one is 3? You can solve for the values of x and y respectively in each case, and see if it contradicts the condition "... whole numbers x and y between 1 and 10 inclusive ...".
Please tell me if you are still stuck.