document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199522>Question 1199522</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32%2F9+=+a+%2B+1%2F%28%28c%2B1%29%2Fb%29 ALIGN=MIDDLE  ALT=\"32%2F9+=+a+%2B+1%2F%28%28c%2B1%29%2Fb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where a,b and c are positive integers, and b < c, evaluate the smallest possible value of abc </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833453 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=833453\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The variable 'a' is a positive integer {1,2,3,...}<BR>\n" );
document.write( "The smallest possible value is a = 1.\r<BR>\n" );
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document.write( "If a = 1, then,<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32%2F9+=+1+%2B+1%2F%28%28c%2B1%29%2Fb%29 ALIGN=MIDDLE  ALT=\"32%2F9+=+1+%2B+1%2F%28%28c%2B1%29%2Fb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32%2F9+-+1+=+1%2F%28%28c%2B1%29%2Fb%29 ALIGN=MIDDLE  ALT=\"32%2F9+-+1+=+1%2F%28%28c%2B1%29%2Fb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32%2F9+-+9%2F9+=+1%2F%28%28c%2B1%29%2Fb%29 ALIGN=MIDDLE  ALT=\"32%2F9+-+9%2F9+=+1%2F%28%28c%2B1%29%2Fb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23%2F9+=+1%2F%28%28c%2B1%29%2Fb%29 ALIGN=MIDDLE  ALT=\"23%2F9+=+1%2F%28%28c%2B1%29%2Fb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23%2F9+=+b%2F%28c%2B1%29 ALIGN=MIDDLE  ALT=\"23%2F9+=+b%2F%28c%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "There are infinitely many solution pairings for b,c<BR>\n" );
document.write( "Here's a small subset of possible integer solutions<BR>\n" );
document.write( "b = 23, c = 8<BR>\n" );
document.write( "b = 46, c = 17<BR>\n" );
document.write( "b = 69, c = 26<BR>\n" );
document.write( "b = 92, c = 35<BR>\n" );
document.write( "The values of b are multiples of 23. The values of c are one less than a multiple of 9.\r<BR>\n" );
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document.write( "Unfortunately all of these b,c pairings do not satisfy the condition that b < c. <BR>\n" );
document.write( "We'll ignore this value of 'a'.\r<BR>\n" );
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document.write( "If a = 2, then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32%2F9+=+a+%2B+1%2F%28%28c%2B1%29%2Fb%29 ALIGN=MIDDLE  ALT=\"32%2F9+=+a+%2B+1%2F%28%28c%2B1%29%2Fb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "leads to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14%2F9+=+b%2F%28c%2B1%29 ALIGN=MIDDLE  ALT=\"14%2F9+=+b%2F%28c%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "follow similar steps shown in the previous section.\r<BR>\n" );
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document.write( "This time we have this subset of solutions<BR>\n" );
document.write( "b = 14, c = 8<BR>\n" );
document.write( "b = 28, c = 17<BR>\n" );
document.write( "b = 42, c = 26<BR>\n" );
document.write( "b = 56, c = 35<BR>\n" );
document.write( "Like before the condition b < c isn't met.<BR>\n" );
document.write( "We'll ignore this value of 'a'.\r<BR>\n" );
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document.write( "If a = 3, then <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32%2F9+=+a+%2B+1%2F%28%28c%2B1%29%2Fb%29 ALIGN=MIDDLE  ALT=\"32%2F9+=+a+%2B+1%2F%28%28c%2B1%29%2Fb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "leads to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F9+=+b%2F%28c%2B1%29 ALIGN=MIDDLE  ALT=\"5%2F9+=+b%2F%28c%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then we have this subset of solutions (out of infinitely many).<BR>\n" );
document.write( "b = 5, c = 8<BR>\n" );
document.write( "b = 10, c = 17<BR>\n" );
document.write( "b = 15, c = 26<BR>\n" );
document.write( "b = 20, c = 35<BR>\n" );
document.write( "Finally the condition b < c has been met.\r<BR>\n" );
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document.write( "Go for the smallest b,c pairing so that we arrive at the smallest product.<BR>\n" );
document.write( "a*b*c = 3*5*8 = 120\r<BR>\n" );
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document.write( "If a = 4, then <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32%2F9+=+a+%2B+1%2F%28%28c%2B1%29%2Fb%29 ALIGN=MIDDLE  ALT=\"32%2F9+=+a+%2B+1%2F%28%28c%2B1%29%2Fb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "leads to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F9+=+b%2F%28c%2B1%29 ALIGN=MIDDLE  ALT=\"-4%2F9+=+b%2F%28c%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then that can be arranged into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F9+=+b%2F%28c%2B1%29 ALIGN=MIDDLE  ALT=\"-4%2F9+=+b%2F%28c%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%28c%2B1%29+=+9b ALIGN=MIDDLE  ALT=\"-4%28c%2B1%29+=+9b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b > 0 so 9b > 0 as well. The right hand side is positive.\r<BR>\n" );
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document.write( "But c > 0 leads to c+1 > 0 and -4(c+1) < 0<BR>\n" );
document.write( "The left hand side is negative.\r<BR>\n" );
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document.write( "In short, we do not have any positive integer solutions when a = 4, b > 0 and c > 0.<BR>\n" );
document.write( "We would need to allow b or c to be negative if we wanted solutions.\r<BR>\n" );
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document.write( "Similar situations happen when a = 5, a = 6, etc.<BR>\n" );
document.write( "It allows us to ignore these values of 'a'.\r<BR>\n" );
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document.write( "Notice how <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32%2F9+=+3+%2B+5%2F9 ALIGN=MIDDLE  ALT=\"32%2F9+=+3+%2B+5%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > where 3 is the whole number part. <BR>\n" );
document.write( "This is why a > 3 leads to b or c being negative. \r<BR>\n" );
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document.write( "Answer: <font color=red>120</font><BR>\n" );
document.write( "This minimized product occurs when (a,b,c) = (3,5,8).<BR>\n" );
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