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Find the 3-Digit Values A, B and C


Date: 08/08/99 at 22:03:30
From: Brad Goorman
Subject: Stuck on a problem

Dear Dr. Math,

I have a math problem that I am stuck on. Where do I start?

Q: The digits of the 3-digit integers a, b, and c are the nine nonzero 
digits 1, 2, 3, ..., 9, each of them appearing exactly once. Given 
that the ratio a:b:c is 1:3:5, determine a, b and c.

Sincerely,

Brad Goorman


Date: 08/08/99 at 22:50:38
From: Doctor Ian
Subject: Re: Stuck on a problem

Hi Brad,

Whenever you don't know what else to do, start by writing down what 
you know. You're going to have three 3-digit numbers whose values are 
in a particular ratio:

     a : b : c   =   _ _ _ : _ _ _ : _ _ _   =   1 : 3 : 5


When you've done that, start looking for ways to rule out bogus 
solutions. For example, you know that if a ends in 1, then b must end 
in 3 and c must end in 5.  Try all the other possibilities:

     _ _ 1 : _ _ 3 : _ _ 5
         2       6       0     0 can't appear
         3       9       5
         4       2       0     0 can't appear
         5       5       5     5 can't appear three times 
         6       8       0     0 can't appear
         7       1       5
         8       4       0     0 can't appear
         9       7       5

In other words, the only possible combinations for the final digits 
are:

     _ _ 1 : _ _ 3 : _ _ 5
         3       9       5
         7       1       5
         9       7       5

You can apply similar constraints to the initial digits. In fact, the 
initial digit of a can only be 1 (do you see why?), so this rules 
out two of the four possibilities:

  
     1 _ 3 : _ _ 9 : _ _ 5
     1   9       7       5

And the initial digit of b can only be 3 or 4. Can you take it from 
here?

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