document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=152422>Question 152422</A>: <I><SPAN STYLE=\"word-break: break-all;\"> SUPPOSE THAT YOU HAVE AN UNLIMITED SUPPLY OF 5 CENT STAMPS AND 11 CENT STAMPS.  YOU CAN MAKE EXACT POSTAGE OF 21 CENT WITH 2-5 CENT STAMPS AND 1-11 CENT STAMP.  hOWEVER YOU CANNOT MAKE EXACT POSTAGE OF 23 CENTS BY USING ONLY 5 CENT STAMPS AND 11 CENT STAMPS.  WHAT IS THE GREATEST AMOUNT OF EXACT POSTAGE YOU CANNOT MAKE BY USING THESE STAMPS? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #112045 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=112045\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> An interesting problem to think about and to use in developing an awareness of numbers.<BR>\n" );
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document.write( "Let's begin by thinking about the first 10 numbers ... 1 through 10<BR>\n" );
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document.write( "In this group of 10 numbers, by using 5 and 11 you can only make the numbers 5 (one 5 cent<BR>\n" );
document.write( "stamp) and 10 (two 5 cent stamps). If you think about it, you will now be able to make any<BR>\n" );
document.write( "number that ends in 5 or 0 by just adding a series of 10 (two fives) to the number 5 <BR>\n" );
document.write( "(one 5 cent stamp) or a series of 10 to the number 10.  For example, 15 can be made by<BR>\n" );
document.write( "adding two 5 cent stamps to a 5, 25 can be made by adding four 5 cents stamps to a 5,<BR>\n" );
document.write( "35 can be made by adding six 5 cent stamps to a 5, and so on.  Similarly, you can make<BR>\n" );
document.write( "any number that ends in a zero, by adding a series of 2 fives (that is 10 cents) to<BR>\n" );
document.write( "two fives ... 20 would be two 5 cent stamps plus two more 5 cent stamps; 30 would be<BR>\n" );
document.write( "two 5 cent stamps plus four more 5 cent stamps, forty would be two 5 cent stamps plus <BR>\n" );
document.write( "six more 5 cent stamps; and so on.<BR>\n" );
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document.write( "Now let's think about the next 10 numbers ... 11 through 20<BR>\n" );
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document.write( "In this group of 10 numbers, by using 5 and 11 you can make the numbers 11 (one 11 cent <BR>\n" );
document.write( "stamp); 15 (three 5 cent stamps); 16 (one 11 cent stamp and one 5 cent stamp); and 20 <BR>\n" );
document.write( "(four 5 cent stamps). So in this group of 10 numbers we have 11, 15, 16, and 20. Using our<BR>\n" );
document.write( "previous method of just adding groups of 10 to these numbers, we can now extend this series<BR>\n" );
document.write( "to any number that ends in 1, 5, 6, or 0.  As examples, 21 would be an 11 cent stamp and <BR>\n" );
document.write( "two 5 cent stamps, 31 would be an 11 cent stamp and four 5 cent stamps, 41 would be<BR>\n" );
document.write( "an 11 cent stamp and six 5 cent stamps, and so on. Similarly, 26 would be an 11 cent stamp <BR>\n" );
document.write( "plus a 5 cent stamp plus two more 5 cent stamps and 36 would be an 11 cent stamp plus<BR>\n" );
document.write( "a 5 cent stamp plus 4 more 5 cent stamps. And so on.<BR>\n" );
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document.write( "In the next group of 10 numbers ... 21 through 30 ... we have seen that we can represent<BR>\n" );
document.write( "21 (an 11 cent stamp plus two 5 cent stamps) and 25 (five 5-cent stamps) and 26 (two 11 cent <BR>\n" );
document.write( "stamps plus one 5 cent stamp) and 30 (six 5 cent stamps).  But we can also represent<BR>\n" );
document.write( "22 (two 11 cent stamps) and 26 (an 11 cent stamp and three 5 cent stamps) and 27 (two 11 <BR>\n" );
document.write( "cent stamps plus one 5 cent stamp). So in this series of numbers we can represent <BR>\n" );
document.write( "21, 22, 25, 26, 27, and 30. Therefore, by adding pairs of 5 cent stamps to these numbers <BR>\n" );
document.write( "we can now represent any number ending in 1, 2, 5, 6, 7, and 0.<BR>\n" );
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document.write( "In the series of 31 through 40 we will add the capability of 33 (three 11 cent stamps)<BR>\n" );
document.write( "and 38 (three 11 cent stamps and a 5 cent stamp). This means the series from 31 to 40 <BR>\n" );
document.write( "not only contains the numbers 31, 32, 35, 36, 37, and 40 but it also contains the numbers<BR>\n" );
document.write( "33 and 38. So this series consists of 31, 32, 33, 35, 36, 37, 38, and 40. It is missing the<BR>\n" );
document.write( "two numbers 34 and 39.<BR>\n" );
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document.write( "Now let's go to the numbers in the series 41 through 50. By adding two 5 cent stamps to each<BR>\n" );
document.write( "of the numbers in the series 31 through 40, we can see that we can represent 41, 42, 43,<BR>\n" );
document.write( "45, 46, 47, 48, and 50.  But we can also represent 44 with four 11 cent stamps and 49<BR>\n" );
document.write( "with four 11 cent stamps plus a 5 cent stamp.  Therefore, in this series we can represent all<BR>\n" );
document.write( "the numbers ... 41, 42, 43, 44, 45, 46, 47, 48, 49, and 50.<BR>\n" );
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document.write( "From that point on we can represent any number by just adding some number of 10 cents (two 5 <BR>\n" );
document.write( "cent stamps) to each number in the series 41 through 50.  Want to represent 1008? Just <BR>\n" );
document.write( "recognize that this number ends in 8 so it is the representation of 48 (four 11 cent stamps <BR>\n" );
document.write( "plus a 5<BR>\n" );
document.write( "cent stamp for a total of 48 cents) and you add to that 48 cents 960 cents more (192 five <BR>\n" );
document.write( "cent stamps) to get 1008 cents of postage.<BR>\n" );
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document.write( "So the biggest number that we could not represent using a combination of 11 cent and<BR>\n" );
document.write( "5 cent stamps was 39.<BR>\n" );
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document.write( "Hope this helps you to understand the problem a little better. If you think about this<BR>\n" );
document.write( "a little it will probably begin to make some sense. It's easier to do than it is to <BR>\n" );
document.write( "explain it, that's for sure.<BR>\n" );
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