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### Find the 3-Digit Values A, B and C

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Date: 08/08/99 at 22:03:30
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,

```

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

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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