document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190206>Question 1190206</A>: <I><SPAN STYLE=\"word-break: break-all;\"> When a number is divided by 9, and the remainder is 6. What is the remainder when six times this number is divided by 9. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #821799 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=821799\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "x = some unknown integer<BR>\n" );
document.write( "x/9 = quotient + (remainder/9)<BR>\n" );
document.write( "x/9 = q + (r/9)<BR>\n" );
document.write( "x = 9q + r<BR>\n" );
document.write( "The items q and r will be integers as well.<BR>\n" );
document.write( "The r can only take on integers from the set {0, 1, 2, 3, 4, 5, 6, 7, 8}\r<BR>\n" );
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document.write( "Dividing this unknown number (x) over 9 leads to a remainder 6, so r = 6<BR>\n" );
document.write( "x = 9q + r<BR>\n" );
document.write( "x = 9q + 6<BR>\n" );
document.write( "x = 3(3q + 2)\r<BR>\n" );
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document.write( "Now multiply both sides by 6<BR>\n" );
document.write( "x = 3(3q + 2)<BR>\n" );
document.write( "6x = 6*3(3q + 2)<BR>\n" );
document.write( "6x = 18(3q + 2)<BR>\n" );
document.write( "6x = 9*2(3q + 2)\r<BR>\n" );
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document.write( "Dividing both sides by 9 gets us<BR>\n" );
document.write( "6x = 9*2(3q + 2)<BR>\n" );
document.write( "6x/9 = 9*2(3q + 2)/9<BR>\n" );
document.write( "6x/9 = 2(3q + 2)<BR>\n" );
document.write( "Notice the 9's cancel on the right hand side. <BR>\n" );
document.write( "We have some integer that represents the quotient with zero remainder. \r<BR>\n" );
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document.write( "A numeric example:\r<BR>\n" );
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document.write( "The number x = 72 is a multiple of 9 because 8*9 = 72.<BR>\n" );
document.write( "Add on 6 more to guarantee we have remainder 6<BR>\n" );
document.write( "72+6 = 78\r<BR>\n" );
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document.write( "x/9 = 78/9 = 8 remainder 6\r<BR>\n" );
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document.write( "So 6x = 6*78 = 468<BR>\n" );
document.write( "And 6x/9 = 468/9 = 52 remainder 0\r<BR>\n" );
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document.write( "Answer: remainder = 0<BR>\n" );
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