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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?

 Apr 19, 2022
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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.

 Apr 19, 2022

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