Team's Final ScoreDate: 10 Jul 1995 11:38:13 -0400 From: Anonymous Subject: addition Question: The rules of a certain game allow a team to score either 3 points or 8 points. A team's final score will be any combination of these points. For example, 14=3+3+8 and 27=8+8+8+3 Which numbers cannot be a team's final score? From: Michelle Arseneau Date: 11 Jul 1995 21:29:57 -0400 From: Dr. Ken Subject: Re: addition Hello there! This is a great question. To start out, we could draw a number line and eliminate all the numbers that we know we _can_ get. My number line here will be a bunch of $'s: $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$... (there are 50 of them) 1234567... First off, let's eliminate all the multiples of three: $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$... Now let's eliminate all the multiples of eight. I'll put apostrophes in them for clarity's sake: $$ $$ $' $$ $$ '$ $$ $$ $$ $$ $' $$ $$ '$ $$ $$ $$... Now notice something. Everywhere there's an apostrophe, we can add threes to that place value and eliminate more numbers. I'll do it in two steps and fill the slots with a , : $$ $$ $' $, $, ', $, $, $, $, $' $, $, ', $, $, $, $$ $$ $' $, $, ', ,, ,, ,, ,, ,' ,, ,, ', ,, ,, ,, Hey! There's not much left. The first pass took out all the big numbers that have a remainder of 2 when divided by 3, and the second took out all the big numbers that have a remainder of 1 when divided by 3. Since all the numbers that have a remainder of 0 when divided by 3 were already gone, there are no numbers past 13 left (all numbers have a remainder of 0, 1, or 2 when divided by 3). So by looking at the diagram, we have our answer: we could never get a score of 1,2,4,5,7,10, or 13: $$ $$ $' $, $, ', ,, ,, ,, ,, ,' ,, ,, ', ,, ,, ,, 12 45 7 10 13 Thanks for the question! -K |
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