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/
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