document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=29415>Question 29415</A>: <I><SPAN STYLE=\"word-break: break-all;\"> show that 3 is a factor of n^3+2n for all positive integers n. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #16260 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=16260\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> show that 3 is a factor of n^3+2n for all positive integers n<BR>\n" );
document.write( "HOPE YOU WERE TAUGHT FERMAT'S THEOREM....IT SAYS IF P IS A PRIME NUMBER THEN <BR>\n" );
document.write( "N^P=N(MOD P)...SINCE 3 IS A PRIME NUMBER WE HAVE<BR>\n" );
document.write( "N^3=N(MOD 3)<BR>\n" );
document.write( "SO 3 DIVIDES N^3-N<BR>\n" );
document.write( "SO 3 DIVIDES N^3-N+3N SINCE 3N IS DIVISIBLE BY 3<BR>\n" );
document.write( "SO 3 DIVIDES N^3+2N.....<BR>\n" );
document.write( "IF YOU ARE NOT TAUGHT THIS THEOREM ..COME BACK...I SHALL GIVE ALTERNATIVE...<BR>\n" );
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document.write( "ok ,let us prove it without using fermats theorem<BR>\n" );
document.write( "SINCE WE HAVE THE NUMBER AS 3 WHICH IS VERY SMALL ,WE NEED NOT GO FOR<BR>\n" );
document.write( "COMPLICATED METHODS ,BUT CAN USE A SIMPLER METHOD.INDUCTION CAN ALSO<BR>\n" );
document.write( "BE USED.<BR>\n" );
document.write( "LET US SEE<BR>\n" );
document.write( "METHOD 1.....<BR>\n" );
document.write( "T.S.T .....N^3+2N IS DIVISIBLE BY 3....<BR>\n" );
document.write( "LET US PUT ...........Z=N^3+2N<BR>\n" );
document.write( "WHEN A NUMBER IS DIVIDED BY 3 WE CAN HAVE 3 POSSIBLITIES ON REMAINDER...<BR>\n" );
document.write( "1.REMAINDER IS ZERO ...THAT IS THE NUMBER IS A MULTIPLE OF 3 ...N = 3<BR>\n" );
document.write( "X WHERE X IS AN INTEGER..SO<BR>\n" );
document.write( "Z=N^3 +2N = (3X)^3+2*3X= 27X^3+6X=3(9X^2+2X) = 3*M<BR>\n" );
document.write( "WHERE M = 9X^2+2X IS AN INTEGER...SINCE X IS AN INTEGER .<BR>\n" );
document.write( "HENCE Z IS DIVISIBLE BY 3.<BR>\n" );
document.write( "2.REMAINDER IS ...1...THAT IS THE NUMBER  N =3X+1 WHERE X IS AN INTEGER.<BR>\n" );
document.write( "Z=(3X+1)^3+2(3X+1) = 27X^3+27X^2+9X+1+6X+2<BR>\n" );
document.write( "=3(9X^3+9X^2+5X+1)=3*M WHERE<BR>\n" );
document.write( "M=9X^3+9X^2+5X+1 IS AN INTEGER SINCE X IS AN INTEGER.<BR>\n" );
document.write( "HENCE Z IS DIVISIBLE BY 3.\r<BR>\n" );
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document.write( "3.REMAINDER IS ...2..THAT IS THE NUMBER  N =3X+2 WHERE X IS AN INTEGER.<BR>\n" );
document.write( "Z=(3X+2)^3+2(3X+2) = 27X^3+54X^2+36X+8+6X+4<BR>\n" );
document.write( "=3(9X^3+18X^2+12X+4)=3*M WHERE<BR>\n" );
document.write( "M=9X^3+18X^2+12X+4 IS AN INTEGER SINCE X IS AN INTEGER.<BR>\n" );
document.write( "HENCE Z IS DIVISIBLE BY 3.<BR>\n" );
document.write( "HENCE Z IS ALWAYS  DIVISIBLE BY 3 FOR ANY INTEGRAL VALUE OF N.\r<BR>\n" );
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document.write( "INDUCTION METHOD OF PROOF....<BR>\n" );
document.write( "IN INDUCTION WE FOLLOW 4 STEPS...\r<BR>\n" );
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document.write( "I STEP...WE CHECK WHETHER THE THEOREM IS TO BE PROVED TRUE FOR  ALL<BR>\n" );
document.write( "NATURAL NUMBERS  N...<BR>\n" );
document.write( "YES ...SO...OK...\r<BR>\n" );
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document.write( "II STEP..WE CHECK WHETHER THE GIVEN STATEMENT IS TRUE FOR N=1<BR>\n" );
document.write( "ZN=N^3+2N IS DIVISIBLE BY 3 IS THE THEOREM..SO FOR<BR>\n" );
document.write( "N = 1 WE HAVE<BR>\n" );
document.write( "Z1=1^3+2*1=3...IS DIVISIBLE BY 3 ...OK.\r<BR>\n" );
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document.write( "III STEP...WE ASSUME IT IS TRUE FOR N=K AN INTEGER...THAT IS...<BR>\n" );
document.write( "ZK = K^3+2K.. IS DIVISIBLE BY 3...ASSUMED..<BR>\n" );
document.write( "THAT IS ...ZK = K^3+2K=3M... WHERE M IS AN INTEGER.\r<BR>\n" );
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document.write( "IV STEP...IF III IS TRUE THEN ,WE TRY TO PROVE THAT IT IS ALSO TRUE<BR>\n" );
document.write( "FOR N=K+1...THAT IS TO PROVE THAT<BR>\n" );
document.write( "Z(K+1)=(K+1)^3+2(K+1) IS DIVISIBLE BY 3..<BR>\n" );
document.write( "Z(K+1)=(K+1)^3+2(K+1)=K^3+3K^2+3K+1+2K+2<BR>\n" );
document.write( "=(K^3+2K)+3(K^2+K+1)=3M+3*INTEGER ....SINCE K^2+K+1 IS AN INTEGER,K<BR>\n" );
document.write( "BEING AN INTEGER...HENCE<BR>\n" );
document.write( "Z(K+1)=3*(M+INTEGER )= 3 * INTEGER<BR>\n" );
document.write( "THAT IS Z(K+1) IS DIVISIBLE BY 3 IF ZK IS DIVISIBLE BY3\r<BR>\n" );
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document.write( "CONCLUDING ARGUMENT....<BR>\n" );
document.write( "WE PROVED IN II STEP THAT THE THEOREM IS TRUE FOR K=1..BY IV STEP,THEN<BR>\n" );
document.write( "IT IS TRUE FOR K+1 =1+1 =2...THEN IT IS TRUE FOR K+1=2+1=3......SO ON<BR>\n" );
document.write( "...HENCE IT IS TRUE FOR ALL NATURAL NUMBERS N...\r<BR>\n" );
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document.write( "> ok ,let us prove it without using fermats theorem<BR>\n" );
document.write( "> SINCE WE HAVE THE NUMBER AS 3 WHICH IS VERY SMALL ,WE NEED NOT GO FOR<BR>\n" );
document.write( "> COMPLICATED METHODS ,BUT CAN USE A SIMPLER METHOD.INDUCTION CAN ALSO<BR>\n" );
document.write( "> BE USED.<BR>\n" );
document.write( "> LET US SEE<BR>\n" );
document.write( "> METHOD 1.....<BR>\n" );
document.write( "> T.S.T .....N^3+2N IS DIVISIBLE BY 3....<BR>\n" );
document.write( "> LET US PUT ...........Z=N^3+2N<BR>\n" );
document.write( "> WHEN A NUMBER IS DIVIDED BY 3 WE CAN HAVE 3 POSSIBLITIES ON REMAINDER...<BR>\n" );
document.write( "> 1.REMAINDER IS ZERO ...THAT IS THE NUMBER IS A MULTIPLE OF 3 ...N = 3<BR>\n" );
document.write( "> X WHERE X IS AN INTEGER..SO<BR>\n" );
document.write( "> Z=N^3 +2N = (3X)^3+2*3X= 27X^3+6X=3(9X^2+2X) = 3*M<BR>\n" );
document.write( "> WHERE M = 9X^2+2X IS AN INTEGER...SINCE X IS AN INTEGER .<BR>\n" );
document.write( "> HENCE Z IS DIVISIBLE BY 3.<BR>\n" );
document.write( "> 2.REMAINDER IS ...1...THAT IS THE NUMBER  N =3X+1 WHERE X IS AN INTEGER.<BR>\n" );
document.write( "> Z=(3X+1)^3+2(3X+1) = 27X^3+27X^2+9X+1+6X+2<BR>\n" );
document.write( "> =3(9X^3+9X^2+5X+1)=3*M WHERE<BR>\n" );
document.write( "> M=9X^3+9X^2+5X+1 IS AN INTEGER SINCE X IS AN INTEGER.<BR>\n" );
document.write( "> HENCE Z IS DIVISIBLE BY 3.<BR>\n" );
document.write( "><BR>\n" );
document.write( "> 3.REMAINDER IS ...2..THAT IS THE NUMBER  N =3X+2 WHERE X IS AN INTEGER.<BR>\n" );
document.write( "> Z=(3X+2)^3+2(3X+2) = 27X^3+54X^2+36X+8+6X+4<BR>\n" );
document.write( "> =3(9X^3+18X^2+12X+4)=3*M WHERE<BR>\n" );
document.write( "> M=9X^3+18X^2+12X+4 IS AN INTEGER SINCE X IS AN INTEGER.<BR>\n" );
document.write( "> HENCE Z IS DIVISIBLE BY 3.<BR>\n" );
document.write( "> HENCE Z IS ALWAYS  DIVISIBLE BY 3 FOR ANY INTEGRAL VALUE OF N.<BR>\n" );
document.write( "> I SHALL GIVE YOU SEPERATE MAIL ON INDUCTION..<BR>\n" );
document.write( "> VENUGOPAL.<BR>\n" );
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document.write( "> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );