document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1099272>Question 1099272</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many different integers can be represented as a sum of four distinct numbers chosen from the set ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #713736 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=713736\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br>Tutor Edwin has, as usual, provided an excellent solution to your problem.<br><BR>\n" );
document.write( "I would approach the problem a bit differently; so let me show you how I would solve the problem.<br><BR>\n" );
document.write( "You might find one solution method or the other more to your liking; I'm not saying one method is better than the other....<br><BR>\n" );
document.write( "The numbers are<BR>\n" );
document.write( "6, 17, 28, ..., 83, 94, 105<br><BR>\n" );
document.write( "They form an arithmetic sequence with common difference 11.<br><BR>\n" );
document.write( "The smallest sum possible from 4 of these number is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2B17%2B28%2B39+=+90 ALIGN=MIDDLE  ALT=\"6%2B17%2B28%2B39+=+90\"   DISABLED_style= \"max-width:90%; height:auto;\" >;<BR>\n" );
document.write( "the largest sum possible is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72%2B83%2B94%2B105+=+354 ALIGN=MIDDLE  ALT=\"72%2B83%2B94%2B105+=+354\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Because the difference between terms of the sequence is a constant 11, the only sums possible using 4 of the numbers are the numbers that are some multiple of 11 larger than 90 and less than or equal to 354.<br><BR>\n" );
document.write( "That is, the sums that can be formed are the numbers in the arithmetic sequence<BR>\n" );
document.write( "90, 101, 112, ..., 332, 343, 354<br><BR>\n" );
document.write( "There is a simple standard calculation for finding the number of numbers in that sequence -- (last term minus first term); divided by the common difference; plus 1:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=354-90+=+264 ALIGN=MIDDLE  ALT=\"354-90+=+264\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=264%2F11+=+24 ALIGN=MIDDLE  ALT=\"264%2F11+=+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%2B1+=+25 ALIGN=MIDDLE  ALT=\"24%2B1+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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