Greatest Common Factor

Date: 06/02/99 at 23:18:13
From: christopher west
Subject: Greatest common factor

The GCF of two numbers is 479. One number is even, and the other 
number is odd. One number is NOT a multiple of the other. What are 
the smallest numbers these could be?  

I have used calculator, paper and pencil, and mental math, and still 
cannot figure it out.

Date: 06/03/99 at 08:35:41
From: Doctor Peterson
Subject: Re: Greatest common factor

Hi, Christopher.

Let's take a simpler problem, and see what reasoning we can use:

    The GCF of two numbers is 7. One number is even, and the other
    number is odd. One number is NOT a multiple of the other. What are 
    the smallest numbers these could be?

What we're told is that the two numbers A and B have these properties:

    The GCF of A and B is 7.
    A is even (so it is a multiple of 2 as well as of 7).
    B is odd (so it is not a multiple of 2).
    A is not a multiple of B and B is not a multiple of A.

(This last statement could be a misinterpretation; a popular trick 
question involves the statement that "one of the two coins is not a 
nickel," and the solution requires taking this to mean that one is 
not, but the other IS a nickel. I'm assuming this is not a trick 
question where B is not a multiple of A, but A IS a multiple of B.)

We have to start with 7 and multiply it by different things to get A 
and B. We know we want to multiply by 2 to get A:

    A = 7 * 2 * ?
    B = 7 * ?

If we just stopped there with A = 14 and B = 7, one would be a 
multiple of the other. So we'll have to multiply B by something other 
than 2, so it won't divide A evenly. What's the smallest number that 
could be?

Now try doing the same thing with your original problem.

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