document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=96633>Question 96633</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are less than 100 balls in a box. If you remove the balls in threes, 1 ball will remain in the box. If you remove the balls in fours, 2 balls will remain. If you remove the balls in fives, 3 balls will remain in the box. If your remove the balls in sixs, four balls will remain.<BR>\n" );
document.write( "How many balls are there in the box? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #70343 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=70343\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The answer is 58 balls are in the box.<BR>\n" );
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document.write( "Probably the easiest way to do this problem is by using intuition to eliminate possible answers.<BR>\n" );
document.write( "The problem tells you that there are 99 possible answers (1 through 99) and one way to find<BR>\n" );
document.write( "the answer is to start with 99 and find out whether you can divide it by 6 and have a remainder<BR>\n" );
document.write( "of 4. If that works, then divide 99 by 5 and see if the remainder is 3. If that works, then<BR>\n" );
document.write( "divide 99 by 4 and see if the remainder is 3, and if that works, divide 99 by 3 and see if<BR>\n" );
document.write( "the remainder is 1. <BR>\n" );
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document.write( "That sure seems like a lot of work. So let's see how we might trim it down.<BR>\n" );
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document.write( "Let's begin by working in reverse. Let's start with the number of 6.  If we take multiples of 6<BR>\n" );
document.write( "and add 4 to each of them, we get 15 possible answers:<BR>\n" );
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document.write( "6 times 1 = 6 + 4 = 10<BR>\n" );
document.write( "6 times 2 = 12 + 4 = 16<BR>\n" );
document.write( "6 times 3 = 18 + 4 = 22<BR>\n" );
document.write( "6 times 4 = 24 + 4 = 28<BR>\n" );
document.write( "6 times 5 = 30 + 4 = 34<BR>\n" );
document.write( "6 times 6 = 36 + 4 = 40<BR>\n" );
document.write( "6 times 7 = 42 + 4 = 46<BR>\n" );
document.write( "6 times 8 = 48 + 4 = 52<BR>\n" );
document.write( "6 times 9 = 54 + 4 = 58<BR>\n" );
document.write( "6 times 10 = 60 + 4 = 64<BR>\n" );
document.write( "6 times 11 = 66 + 4 = 70<BR>\n" );
document.write( "6 times 12 = 72 + 4 = 76<BR>\n" );
document.write( "6 times 13 = 78 + 4 = 82<BR>\n" );
document.write( "6 times 14 = 84 + 4 = 88<BR>\n" );
document.write( "6 times 15 = 90 + 4 = 94<BR>\n" );
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document.write( "The right side of this table shows those numbers that if you divide them by 6 the remainder<BR>\n" );
document.write( "is 4. So there are only 15 numbers that can be answers to this problem.<BR>\n" );
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document.write( "Next let's look at the possible answers insofar as 5 is involved. Multiples of 5 end in only<BR>\n" );
document.write( "two numbers - zero and 5. When you add 3 (the remainder) to these possibilities, the answer<BR>\n" );
document.write( "will either end in 3 or in 8.  Now you look down the table for the sixes, and you find<BR>\n" );
document.write( "that of the possible answers for the sixes none of them end in 3 and only three end in 8 <BR>\n" );
document.write( "... 28, 58, and 88. So only 28, 58, and 88 will satisfy the requirements that if you divide <BR>\n" );
document.write( "them by 6 you will get a remainder of 4 AND if you divide them by 5 the remainder will be 3.<BR>\n" );
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document.write( "Next let's check these three numbers by seeing if they satisfy the requirement that if they<BR>\n" );
document.write( "are divided by 4 the remainder is 2.  Divide 28 by 4 and the answer is 7 with no remainder. <BR>\n" );
document.write( "Divide 58 by 4 and the answer is 14 with a remainder of 2. Finally, divide 88 by 4 and the<BR>\n" );
document.write( "answer is 22 with no remainder. At this point we know that the only possible answer is 58.<BR>\n" );
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document.write( "Check this answer by dividing 58 by 3 and seeing if the remainder is 1.  If you divide 58 by 3<BR>\n" );
document.write( "the answer is 19 with a remainder of 1. So this checks and confirms the answer we found.<BR>\n" );
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document.write( "So we found the answer by finding a way to narrow the number of possibilities down to a manageable<BR>\n" );
document.write( "amount and eliminate a lot of work. <BR>\n" );
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document.write( "Hope this helps you to think of other things that might be done to solve this problem and<BR>\n" );
document.write( "at least to understand that the above possibility exists.<BR>\n" );
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