document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183950>Question 1183950</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that x - y is a factor of x^n - y^n for all positive integers n, using mathematical induction. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814475 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=814475\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Although for the most part the proof of the other tutor is correct, there is one part of the proof which is questionable, <BR>\n" );
document.write( "and it is the part where it is indicated that \r<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E%28k%2B1%29-y%5E%28k%2B1%29=+matrix%283%2C3%2C%22%22%2C%22%22%2C%22%22%2C+%28x%5E%28%28k%2B1%29%2F2%29+-+y%5E%28%28k%2B1%29%2F2%29%29+%2A+%28x%5E%28%28k%2B1%29%2F2%29+%2B+y%5E%28%28k%2B1%29%2F2%29%29%2C%22%22%2C%22%22%2C%22%22%2C%22%22%2C%22%22%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "This is questionable since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k%2B1%29%2F2 ALIGN=MIDDLE  ALT=\"%28k%2B1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT always an integer, and so induction does not apply in that case.  The remedy is to use the inductive hypothesis directly.\r<BR>\n" );
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document.write( "Step 1.  Check for n = 1:  Obviously x - y divides x - y, so staement is true for n = 1.\r<BR>\n" );
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document.write( "Step 2.  (Inductive hypothesis)  Assume true for n = k, i.e., x - y divides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Ek+-+y%5Ek ALIGN=MIDDLE  ALT=\"x%5Ek+-+y%5Ek\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Step 3.  Prove that x - y divides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%28k%2B1%29+-+y%5E%28k%2B1%29 ALIGN=MIDDLE  ALT=\"x%5E%28k%2B1%29+-+y%5E%28k%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%28k%2B1%29+-+y%5E%28k%2B1%29+=+x%5E%28k%2B1%29+-+x%5Ek%2Ay+%2B+x%5Ek%2Ay+-+y%5E%28k%2B1%29+=+x%5Ek%2A%28x-y%29+%2B+y%28x%5Ek+-+y%5Ek%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Since x - y divides itself and by the inductive hypothesis x - y divides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Ek+-+y%5Ek ALIGN=MIDDLE  ALT=\"x%5Ek+-+y%5Ek\"   DISABLED_style= \"max-width:90%; height:auto;\" >, it followes that x - y divides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%28k%2B1%29+-+y%5E%28k%2B1%29 ALIGN=MIDDLE  ALT=\"x%5E%28k%2B1%29+-+y%5E%28k%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Therefore statement is true for <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;\" >, and the statement is PROVED.<BR>\n" );
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