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/
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