Question

What are two 'different' integers with the same sum and multiplication?

Answer 1

Answer:

This is another little essay is an exercise in mathematical recreations; I hope you find it amusing!

When is addition the same as multiplication? In other words, when is the following true?

x + y = x * y

A bit of thinking will help you realize that {x,y} = {2,2} and {0,0} are two solutions to the problem, because:

2 + 2 = 2 * 2 = 4

0 + 0 = 0 * 0 = 0

But are there other solutions? To find out, we need to solve the equation using a little algebra:

x + y = xy

0 = xy - x - y

0 = (x-1)(y-1) - 1 (Confused? See the Postscript below)

1 = (x-1)(y-1)

1/(x-1) = y-1

y = 1/(x-1) + 1

So, there are an infinite number of real number solutions; to find a given y, just compute y = 1/(x-1)+1 (as long as x is not equal to 1; there's no real number solution in that case). For example, the pair {1.5,3} works, because:

1.5 + 3 = 1.5 * 3 = 4.5

Now, let's narrow the question - are there any other real number solutions where x=y, other than 0 and 2? It turns out the answer is no, and here's why, starting with the equation above and changing it since now x=y:

0 = (x-1)(y-1) - 1 (see above)

0 = (x-1)^2 -1 (because x=y now)

1 = (x-1)^2

+1 or -1 = x-1

x = 0 or 2

Now, instead of limiting ourselves to x=y, let's allow any value of x and y, but only if they are both integers. Given that variation, are there any other integer pair solutions? The short answer is no - if you limit yourself to integers, 0 and 2 are all that's possible. Here's why. Since integers are a subset of the real numbers, the equation given above applies:

y = 1/(x-1)+1

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