Headscratch Letter Puzzle

Date: 01/25/2002 at 10:34:21
From: Ting Zi
Subject: Headscratch!

Dear Sir/Madam,

I am stuck on this maths problem. Here is the question:

Headscratch!
You can only use the numbers 0 to 9 for each letter once.

G*B=HA
D*E=E
I*C=C
H*A=DE
F*J=GJ

*=TIMES

Can you please explain how to do this problem?
Ting Zi

Date: 01/25/2002 at 17:01:13
From: Doctor Peterson
Subject: Re: Headscratch!

Hi, Ting.

Could you show me what ideas you had? You may have been close to an 
answer and only made some little mistakes, or you may not have been 
trying the right approach. This takes a little thinking, and I want to 
give you only the hints you need so you can have the experience of 
finding the answer yourself.

The first things I notice are

    D*E=E
    I*C=C

There are only two ways that a number can be multiplied by another 
number and be left unchanged: either the multiplier is 1, or the 
number that stays the same is 0. So there are two possibilities:

    D=1, C=0, so 1*E=E and I*0=0

    I=1, E=0, so D*0=0 and 1*C=C

You can try each of these possibilities and see what happens as you 
continue thinking. When you determine whether there is any solution 
making one assumption, switch to the other assumption and try again.

You can also look at

    F*J=GJ

and make a list of all possible products of this form. You can ignore 
any for which any of the digits are 0 or 1, or where two other than 
the J's are the same (such as 6*2=12). This will leave only four 
possibilities, I believe, which determine the choices for G.

Then look at

    G*B=HA
    H*A=DE

You will have chosen a value for D or E, so you can find all 
possibilities for H*A, and then see which can be factored to give G*B.

If you need more than these hints, write back and show your thinking 
as far as you get, so I can help where you really need help.

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