document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=53236>Question 53236</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find least no in which when divide by 5,7,9 then it leaves remainder 1,2,3 respectively. What is logic for solving this problem. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #35476 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=35476\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find least no in which when divide by 5,7,9 then it leaves remainder 1,2,3 respectively. What is logic for solving this problem.<BR>\n" );
document.write( "I AM NOT SURE HOW FAR YOU STUDIED ON THEORY OF NUMBERS..SO LET ME GIVE YOU A<BR>\n" );
document.write( "COMMONSENSE APPOACH.......AND DOWN BELOW BASED ON THEORY OF NUMBERS<BR>\n" );
document.write( "LET THE NUMBER BE .....N<BR>\n" );
document.write( "N-1|5............N= 5M+1...WHERE M IS AN INTEGER.............1<BR>\n" );
document.write( "N-2|7............5M+1-2|7.....5M-1|7...........5M-1=7P......2<BR>\n" );
document.write( "N-3|9............5M-2|9......5M-2=9Q........................3<BR>\n" );
document.write( "NOW USE EQN.3 BY GIVING Q INCREASING VALUES AND TESTING EQN 2.NOTE THAT P,Q,M,N SHOULD ALL BE INTEGERS.<BR>\n" );
document.write( "I.TRY..Q=2.........5M-2=9*2=18.....M=4....BUT THIS CANT SATISFY EQN2<BR>\n" );
document.write( "NEXT JUMP WILL BE TO Q=2+5=7<BR>\n" );
document.write( "II.TRY.Q=7.........................M=13....NO(NOTE THAT M INCREASES IN STEPS OF 9...FOR Q INCREASIN IN STEPS OF 5)<BR>\n" );
document.write( "III..Q=12..........................M=22.......NO<BR>\n" );
document.write( "IV...Q=17..........................M=31......YES..5*31-1=154|7=22<BR>\n" );
document.write( "SO........... N=5*31+1=156..<BR>\n" );
document.write( "-------------------------------------------------------------------------<BR>\n" );
document.write( "BASED ON CONGRUENCE MODULO<BR>\n" );
document.write( "(5,7)=(5,9)=(7,9)=1 HENCE THERE IS A SOLUTION.<BR>\n" );
document.write( "N = 1(MOD 5)=2(MOD 7) = 3(MOD 9)<BR>\n" );
document.write( "N= 1+5M.....<BR>\n" );
document.write( "1+5M=2(MOD 7)<BR>\n" );
document.write( "5M = 1(MOD 7)....ADD 0 =14(MOD 7) TO BOTH SIDES<BR>\n" );
document.write( "5M = 15 (MOD 7)<BR>\n" );
document.write( "M = 3 (MOD 7)<BR>\n" );
document.write( "M = 7P+3<BR>\n" );
document.write( "N = 1+5(7P+3)=1+35P+15=35P+16<BR>\n" );
document.write( "35P+16 = 3 (MOD 9)<BR>\n" );
document.write( "35P = -13 (MOD 9).......ADD 0=153(MOD 9)<BR>\n" );
document.write( "35P=140(MOD 9)<BR>\n" );
document.write( "P=4(MOD 9)=9Q+4<BR>\n" );
document.write( "N= 35P+16 = 35(9Q+4)+16=315Q+156<BR>\n" );
document.write( "HENCE WE GET DIFFERENT SOLUTIONS FOR THE CONGRUENCIES BY PUTTING Q=0,1,2...ETC<BR>\n" );
document.write( "N= 156,571,786...ETC...<BR>\n" );
document.write( "LEAST VALUEOF N IS 156 AS GOT BOVE.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );