document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=12319>Question 12319</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do u prove this?...Show that 3 divides n^3-n for all positive integers n. And does 4 divide n^4-n for all positive intgers n?  AND i need to use proof by induction i think, i get the first part but then get stuck at the end...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 #6249 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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=6249\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Show that 3 divides n^3-n for all positive integers n. <BR>\n" );
document.write( "Proof: Claim: 3 isa divisor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n%5E3-n+ ALIGN=MIDDLE  ALT=\"+n%5E3-n+\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all integer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+%3E=+1 ALIGN=MIDDLE  ALT=\"n+%3E=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "             .....(**)<BR>\n" );
document.write( "    Basic: When n = 1 , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n%5E3-n+=+1%5E3-+1+=0 ALIGN=MIDDLE  ALT=\"+n%5E3-n+=+1%5E3-+1+=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "           So, (**) is true when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n+=+1 ALIGN=MIDDLE  ALT=\"+n+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    Induction Hypothesis: When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n+=+k ALIGN=MIDDLE  ALT=\"+n+=+k\"   DISABLED_style= \"max-width:90%; height:auto;\" > (**) is true. <BR>\n" );
document.write( "    Hence, there exists an integer q such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+k%5E3-k+=+3q+ ALIGN=MIDDLE  ALT=\"+k%5E3-k+=+3q+\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "    Consider <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28k%2B1%29%5E3-%28k%2B1%29+=+k%5E3+%2B+3%2Ak%5E2+%2B+3%2Ak+%2B+1+-+k+-1 ALIGN=MIDDLE  ALT=\"+%28k%2B1%29%5E3-%28k%2B1%29+=+k%5E3+%2B+3%2Ak%5E2+%2B+3%2Ak+%2B+1+-+k+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "            = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+k%5E3+-k+%2B+3%2Ak%5E2+%2B+3%2Ak++ ALIGN=MIDDLE  ALT=\"+k%5E3+-k+%2B+3%2Ak%5E2+%2B+3%2Ak++\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "            = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Aq+%2B3%2Ak%28k%2B1%29+ ALIGN=MIDDLE  ALT=\"3%2Aq+%2B3%2Ak%28k%2B1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "            = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3%2A%28q%2B+k%5E2%2B+k%29+ ALIGN=MIDDLE  ALT=\"+3%2A%28q%2B+k%5E2%2B+k%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "   This shows 3 is a divisor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28k%2B1%29%5E3-%28k%2B1%29+ ALIGN=MIDDLE  ALT=\"+%28k%2B1%29%5E3-%28k%2B1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "   And so,the inductive proof is complete.  <BR>\n" );
document.write( " <BR>\n" );
document.write( " [Actually,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n%5E3-n ALIGN=MIDDLE  ALT=\"+n%5E3-n\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a product of three consecutive positive integers <BR>\n" );
document.write( " for any positive integer n. Why ? ]\r<BR>\n" );
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document.write( "And does 4 divide n^4-n for all positive intgers n? \r<BR>\n" );
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document.write( " When n =2,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E4-n+=+16+-+2+=+14+ ALIGN=MIDDLE  ALT=\"n%5E4-n+=+16+-+2+=+14+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( " Hence 4 cannot divide n^4-n in general.\r<BR>\n" );
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document.write( " In fact, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E4-n+=+n%2A%28n%5E3+-1%29+=n%2A%28n-1%29%28n%5E2%2Bn%2B1%29+ ALIGN=MIDDLE  ALT=\"n%5E4-n+=+n%2A%28n%5E3+-1%29+=n%2A%28n-1%29%28n%5E2%2Bn%2B1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " Clearly, 2 is a divisor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2A%28n-1%29 ALIGN=MIDDLE  ALT=\"n%2A%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " We only can claim 2 divides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E4-n+ ALIGN=MIDDLE  ALT=\"n%5E4-n+\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all positive n.\r<BR>\n" );
document.write( "\n" );
document.write( " Kenny<BR>\n" );
document.write( " </SPAN>\n" );
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