document.write( "Question 1183950: Show that x - y is a factor of x^n - y^n for all positive integers n, using mathematical induction. \n" ); document.write( "

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Show that x - y is a factor of x^n - y^n for all positive integers n, using mathematical induction.\r
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document.write( "NOTE:   notatation  \"a | b\"  means  \"a divides b\"\r
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document.write( "Base case:  n=1:   (x-y) |  ,  base case holds\r
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document.write( "Hypothesis:  assume  (x-y) |     (*)   NOTE: k>1\r
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document.write( "Step case:  let n=k+1:\r
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document.write( "     RHS becomes  \r
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document.write( "How do we apply (*)?   By reworking this to something that has exponents less than or equal to k:\r
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document.write( "    = 
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document.write( "The left factor is of the form  where  (recall k>1) so we can apply (*) and say  | ()\r
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document.write( "Since (x-y) divides one factor, it divides the entire product (if a|b then a|(bc)) .\r
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document.write( "Proof complete. \n" );
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