document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204315>Question 1204315</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A three-digit number ABC is divided by the two-digit number AC. The quotient is 11 with no remainder. What is the largest possible number ABC? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840491 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=840491\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The three digit number ABC means <BR>\n" );
document.write( "A = hundreds digit<BR>\n" );
document.write( "B = tens digit<BR>\n" );
document.write( "C = units or ones digit\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "ABC is more formally written as 100A+10B+C<BR>\n" );
document.write( "AC becomes 10A+C\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divide those values and we get a quotient of 11 and no remainder.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(100A+10B+C)/(10A+C) = 11<BR>\n" );
document.write( "100A+10B+C = 11(10A+C)<BR>\n" );
document.write( "100A+10B+C = (10+1)(10A+C)<BR>\n" );
document.write( "100A+10B+C = 10(10A+C)+1(10A+C)<BR>\n" );
document.write( "100A+10B+C = 100A+10C+10A+C<BR>\n" );
document.write( "10B+C = 10C+10A+C<BR>\n" );
document.write( "10B = 10C+10A<BR>\n" );
document.write( "0 = 10A-10B+10C<BR>\n" );
document.write( "10(A-B+C) = 0<BR>\n" );
document.write( "A-B+C = 0<BR>\n" );
document.write( "A = B-C\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To make number ABC as large as possible, we need A as large as possible.<BR>\n" );
document.write( "At the same time, we need B to be as large as possible as well.<BR>\n" );
document.write( "For unique single digits B and C, B-C maxes out when these digits are as far away from each other as possible, and when B > C.<BR>\n" );
document.write( "That happens when B = 9 and C = 0<BR>\n" );
document.write( "So A = B-C = 9-0 = 9\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "But A = 9 and B = 9 overlap.<BR>\n" );
document.write( "We assume that A and B are different values. Otherwise the number ABC would be AAC or BBC.<BR>\n" );
document.write( "Let's go for B = 9 and C = 1 instead.<BR>\n" );
document.write( "A = B-C = 9-1 = 8<BR>\n" );
document.write( "This would allow A,B,C to be different integers.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The number ABC = 891 is the largest possible value so that ABC/AC = 11 without a remainder (i.e. remainder is 0).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "More specifically, <BR>\n" );
document.write( "891/81 = 11\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: 891<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );