Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

Coins in a Square Array


Date: 7/8/96 at 9:6:16
From: Anonymous
Subject: Prove that the butler lied

Dear Dr. Math,

A man says to his butler, "I left some valuable coins on the table 
this morning in a square array and now there are only two left."  The 
butler answers, "Three burglars came in, divided the coins equally, 
and left these two because they couldn't split them between them."  

I have to prove that the butler lied.

I know that I can prove it by showing that square numbers when 
divided by 3 do not ever have a remainder of 2, but I don't know how 
to write the formula. I also know that every third square number 
(multiples of 3 will be divisible by 3) won't have a remainder, but I 
don't know how to prove it mathematically.  Can you help?


Date: 7/8/96 at 16:56:5
From: Doctor Pete
Subject: Re: Prove that the butler lied

Clearly, every number leaves a remainder of 0, 1, or 2 when divided by 
3. Suppose n is divisible by 3.  Then n^2 is also divisible by 3.  
Note n+1 leaves a remainder of 1, and (n+1)^2 = n^2 + 2n + 1 will also 
leave a remainder of 1, since both n^2 and 2n are divisible by 3.  
Finally, n+2 leaves a remainder of 2, but (n+2)^2 = n^2 + 4n + 4 
leaves a remainder of 1 again, because 4 leaves a remainder of 1 when 
divided by 3.  Therefore every square leaves a remainder of 0 or 1 
when divided by 3, and never a remainder of 2.  Therefore the butler 
lied.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/