Using Algebra to Find Lucky Numbers

Date: 12/26/2007 at 09:08:14
From: Angela
Subject: a lucky number is a positive integer 19 times the sum of its

A lucky number is a positive integer which is 19 times the sum of its
digits.  How many different lucky numbers are there?

I've just tried guess and check and it hasn't worked.

Date: 12/26/2007 at 16:36:29
From: Doctor Jaffee
Subject: Re: a lucky number is a positive integer 19 times the sum of its

Hi Angela,

This is one of the more interesting problems I've encountered in a
while.  Here is how I approached solving it.  I'm not sure how much
algebra you've had, so if my solution doesn't make any sense, let me
know and I'll try explaining it without using algebra.

First of all, there can't be any 1-digit lucky number because if you
multiply 19 by a 1-digit number you always get something much bigger
than a 1-digit number.

If a lucky number has two digits, let's call the digits x and y and
the value of the number would be 10x + y.  Then 10x + y = 19(x + y). 
But, if you solve this equation for x you get x = -2y, which would
mean that whatever value you pick for y, the value of x would be 
negative.  That can't be, so there are no 2-digit lucky numbers.

But, if a lucky number has 3 digits, x, y, and z, then 100x + 10y + z
= 19(x + y + z).  If you solve this for x, you get x = (y + 2z)/9.  
The smallest that x can be is 1 and that can be accomplished if y = 1
and z = 4.  So, 114 is the smallest lucky number.  1 + 1 + 4 = 6 and 
19 x 6 = 114.

Now, if you add 19 to a lucky number, you will get another lucky 
number (at least, for a while).

Give it a try and if you want to check your answer with me or if you
want some clarification about this problem, write back and we'll 
discuss it some more.

By the way, it can be proven, using a method similar to what I've 
done, that there are no lucky numbers with more than 3 digits. 

Thanks for writing to Dr. Math, and good luck.


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