document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1052975>Question 1052975</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Which number who divided by 7,9,11and remainder 1,2,3? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #668284 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=668284\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x be this unknown number that satisfies all three conditions given. For the sake of simplicity, we'll assume x is positive. \r<BR>\n" );
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document.write( "Use the <a  rel=nofollow HREF=\"http://www-math.ucdenver.edu/~wcherowi/courses/m5410/crt.pdf\">Chinese Remainder Theorem</a>\r<BR>\n" );
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document.write( "I will use the same basic steps and notation as that PDF is showing. \r<BR>\n" );
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document.write( "We have these 3 congruences given<BR>\n" );
document.write( "x = 1 (mod 7)<BR>\n" );
document.write( "x = 2 (mod 9)<BR>\n" );
document.write( "x = 3 (mod 11)\r<BR>\n" );
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document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bi%5D ALIGN=MIDDLE  ALT=\"a%5Bi%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > values are the right hand side values (1, 2 and 3 respectively)  where 'i' refers to the modulus value<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B7%5D+=+1 ALIGN=MIDDLE  ALT=\"a%5B7%5D+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B9%5D+=+2 ALIGN=MIDDLE  ALT=\"a%5B9%5D+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B11%5D+=+3 ALIGN=MIDDLE  ALT=\"a%5B11%5D+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Multiply out the modulus values<BR>\n" );
document.write( "N = 7*9*11 = 693\r<BR>\n" );
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document.write( "Calculate the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%5Bi%5D ALIGN=MIDDLE  ALT=\"M%5Bi%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > values<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%5B7%5D+=+N%2F7+=+693%2F7+=+99 ALIGN=MIDDLE  ALT=\"M%5B7%5D+=+N%2F7+=+693%2F7+=+99\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%5B9%5D+=+N%2F9+=+693%2F9+=+77 ALIGN=MIDDLE  ALT=\"M%5B9%5D+=+N%2F9+=+693%2F9+=+77\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%5B11%5D+=+N%2F11+=+693%2F11+=+63 ALIGN=MIDDLE  ALT=\"M%5B11%5D+=+N%2F11+=+693%2F11+=+63\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then finally we need to compute the multiplicative inverses <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5Bi%5D ALIGN=MIDDLE  ALT=\"y%5Bi%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > values of each <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%5Bi%5D ALIGN=MIDDLE  ALT=\"M%5Bi%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > value\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B7%5D+=+%28M%5B7%5D%29%5E%28-1%29+=+%2899%29%5E%28-1%29+=+1 ALIGN=MIDDLE  ALT=\"y%5B7%5D+=+%28M%5B7%5D%29%5E%28-1%29+=+%2899%29%5E%28-1%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 7)<BR>\n" );
document.write( "Note: 99 = 1 (mod 7) so the solution to 1y = 1 (mod 7) is y = 1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B9%5D+=+%28M%5B9%5D%29%5E%28-1%29+=+%2877%29%5E%28-1%29+=+2 ALIGN=MIDDLE  ALT=\"y%5B9%5D+=+%28M%5B9%5D%29%5E%28-1%29+=+%2877%29%5E%28-1%29+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 9)<BR>\n" );
document.write( "Note: 77 = 5 (mod 9). The solution to 5y = 1 (mod 9) is y = 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B11%5D+=+%28M%5B11%5D%29%5E%28-1%29+=+%2863%29%5E%28-1%29+=+7 ALIGN=MIDDLE  ALT=\"y%5B11%5D+=+%28M%5B11%5D%29%5E%28-1%29+=+%2863%29%5E%28-1%29+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 11)<BR>\n" );
document.write( "Note: 63 = 8 (mod 11) and the solution to 8y = 1 (mod 11) is y = 7\r<BR>\n" );
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document.write( "Now we use the information computed above to get the solution for x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+a%5B7%5D%2AM%5B7%5D%2Ay%5B7%5D+%2B+a%5B9%5D%2AM%5B9%5D%2Ay%5B9%5D+%2B+a%5B11%5D%2AM%5B11%5D%2Ay%5B11%5D ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 693)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2A99%2A1+%2B+2%2A77%2A2+%2B+3%2A63%2A7 ALIGN=MIDDLE  ALT=\"x+=+1%2A99%2A1+%2B+2%2A77%2A2+%2B+3%2A63%2A7\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 693)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1730 ALIGN=MIDDLE  ALT=\"x+=+1730\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 693)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+344 ALIGN=MIDDLE  ALT=\"x+=+344\"   DISABLED_style= \"max-width:90%; height:auto;\" > (mod 693)\r<BR>\n" );
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document.write( "The smallest positive solution for x is <font color=red>344</font>\r<BR>\n" );
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document.write( "The set of all solutions for x is the set of x values such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+693t%2B344 ALIGN=MIDDLE  ALT=\"x+=+693t%2B344\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%3E=0 ALIGN=MIDDLE  ALT=\"t%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer. In other words, <font color=red>x = 344 (mod 693)</font>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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