document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=745014>Question 745014</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if n > 2 ,then show that n^5-5n^3+4n is divisible by 120 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #454744 by </B><A HREF=/tutors/aboutme.mpl?userid=tommyt3rd><B>tommyt3rd(5050)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=tommyt3rd><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/cgi-bin/show-face.mpl?user=tommyt3rd><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=454744\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Clearly we  assume that n is an integer...\r<BR>\n" );
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document.write( "First let's factor:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0An%5E5-5n%5E3%2B4n=n%28n%5E4-5n%5E2%2B4%29=n%28n%5E2-1%29%28n%5E2-4%29=n%28n-1%29%28n%2B1%29%28n-2%29%28n%2B2%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "next we rearrange them is succession:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A%28n-2%29%28n-1%29n%28n%2B1%29%28n%2B2%29%0D%0A ALIGN=MIDDLE  ALT=\"%0D%0A%28n-2%29%28n-1%29n%28n%2B1%29%28n%2B2%29%0D%0A\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which shows that this is a product of 5 consecutive numbers starting with n-2\r<BR>\n" );
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document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A120=12%2A10=2%5E3%2A3%2A5=3%2A5%2A8%0D%0A ALIGN=MIDDLE  ALT=\"%0D%0A120=12%2A10=2%5E3%2A3%2A5=3%2A5%2A8%0D%0A\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now notice that if n>2 the each product is\r<BR>\n" );
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document.write( "1*2*3*4*5<BR>\n" );
document.write( "2*3*4*5*6<BR>\n" );
document.write( "3*4*5*6*7\r<BR>\n" );
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document.write( "and so on\r<BR>\n" );
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document.write( "each of which is a multiple of 3 (every third digit), 5 (every fifth digit), and 8 (there will always be a multiple of 2 and a multiple of 4).\r<BR>\n" );
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document.write( "Therefore they will always divide 120.<BR>\n" );
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