Two Numbers with Equal Sum, Product, and Quotient?

Date: 05/07/2007 at 23:14:15
From: Hannah
Subject: (no subject)

Find two numbers such that when you find the sum, product, and
quotient of the two numbers they are all equal.

I think working with 0 and possibly negative numbers might help, and I
am thinking that the two numbers must be low numbers.  But all of my
tries come up with two out of the three being equal, but the third not
equal.

Date: 05/08/2007 at 10:12:39
From: Doctor Ian
Subject: Re: 

Hi Hannah,

Interesting question.  Presumably you just mean one of the quotients,
right?  For example, if you have the numbers 2 and 3, there are two
possible quotients: 3/2 and 2/3.  

I'd probably start the way you're starting--looking for easy and 
obvious answers, involving small numbers.  

Of course, this is the sort of case where algebra really comes in
handy!  Suppose we call our numbers x and y.  The condition to be
satisfied is

  x + y = x * y = x / y

which translates to three separate conditions, 

  x + y = x * y 

  x + y = x / y

  x * y = x / y

Let's try solving the third equation for x in terms of y:

      x * y = x / y

  x * y * y = x

      y * y = 1

Interesting!  This tells us that if such numbers do exist, one of them
has to be either 1 or -1.  Now we can go back and try the other
equations with those values:

  y = 1:

    x + 1 = x * 1
 
    x + 1 = x / 1


  y = -1:

    x + -1 = x * -1 

    x + -1 = x / -1


Can you find a value that makes each pair of equations true?  Or can
you show that this is impossible?  Right away, the equation

    x + 1 = x * 1

presents a problem:

    x + 1 = x * 1

    x + 1 = x

        1 = 0

So y can't possibly be equal to 1, since that leads to a 
contradiction.  So either y is -1, or there is no solution.

Can you take it from here? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/