document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=363643>Question 363643</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Trying to help my child with homework and can't figure out the formula. Please help:\r<BR>\n" );
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document.write( "There are N number of buttons in a sewing box. <BR>\n" );
document.write( "N is greater then 40.<BR>\n" );
document.write( "N is less then 80.<BR>\n" );
document.write( "N is divisable by 5 with a remainder of 2.<BR>\n" );
document.write( "N is divisable by 7 with a remainder of 4.<BR>\n" );
document.write( "Solve for N.\r<BR>\n" );
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document.write( "We are able to find N by process of elimination (N = 67) but I need the formula to help my child with future similar problems. Please help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #261738 by </B><A HREF=/tutors/aboutme.mpl?userid=Sphinx+pinastri><B>Sphinx pinastri(17)</B></A><img src=\"/images/stars/stars-06.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Sphinx+pinastri><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=Sphinx+pinastri><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=261738\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm not sure that a child will like a general method, but it is<BR>\n" );
document.write( "rather useful in math competitions. Please review a chapter about<BR>\n" );
document.write( "linear Diophantine equations in any math textbook.\r<BR>\n" );
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document.write( "N = 5a + 2 = 7b + 4\r<BR>\n" );
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document.write( "where a and b are some natural numbers. Let's solve this<BR>\n" );
document.write( "Diophantine equation.\r<BR>\n" );
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document.write( "5a - 7b = 2<BR>\n" );
document.write( "5(a - b) = 2(b + 1)\r<BR>\n" );
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document.write( "This can happen when<BR>\n" );
document.write( "a - b = 2<BR>\n" );
document.write( "b + 1 = 5\r<BR>\n" );
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document.write( "i.e. \r<BR>\n" );
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document.write( "a = 6 <BR>\n" );
document.write( "b = 4<BR>\n" );
document.write( "N = 32\r<BR>\n" );
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document.write( "Addition of LCM(7,5)=35 won't change the remainders.<BR>\n" );
document.write( "So the general solution is<BR>\n" );
document.write( "N = 32 + 35*n where n is a natural number.\r<BR>\n" );
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document.write( "Now we need to solve an inequality:<BR>\n" );
document.write( "40 < 32 + 35*n < 80\r<BR>\n" );
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document.write( "i.e.<BR>\n" );
document.write( "n = 1<BR>\n" );
document.write( "N = 67\r<BR>\n" );
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document.write( " </SPAN>\n" );
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