document.write( "Question 773315: use mathematical induction to prove 7^n-3^n is divisible by 4. \n" ); document.write( "
Algebra.Com's Answer #471443 by Edwin McCravy(20056)\"\" \"About 
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use mathematical induction to prove 7^n - 3^n is divisible by 4.
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document.write( "It is true for n = 1 since 71 - 31 = 4 which\r\n" );
document.write( "is divisible by 4.\r\n" );
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document.write( "We also need to show that it is true for n = 2 (which makes this \r\n" );
document.write( "proof a little different from most induction proofs as we will see.)\r\n" );
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document.write( "It is true for n = 2 since 72 - 32 = 49 - 9 = 40 which is\r\n" );
document.write( "divisible by 4.\r\n" );
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document.write( "Assume for some k ≧ 2 , that both 7k - 3k and 7k-1 - 3k-1 are divisible\r\n" );
document.write( "by 4.  Therefore there exist integers p,q such that \r\n" );
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document.write( "(A)   7k - 3k = 4p and \r\n" );
document.write( "(B)   7k-1 - 3k-1 = 4q\r\n" );
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document.write( "Multiply the left side of (A) by (7 + 3) and the right side by 10\r\n" );
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document.write( "(7 + 3)(7k - 3k) = 40p\r\n" );
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document.write( "FOIL the left side\r\n" );
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document.write( "7k+1 - 7·3k + 3·7k - 3k+1 = 40p\r\n" );
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document.write( "Factor 7·3 out of the two middle terms on the left:\r\n" );
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document.write( "7k+1 + 7·3(-3k-1 + 7k-1) - 3k+1 = 40p\r\n" );
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document.write( "Reverse the terms inside the parentheses:\r\n" );
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document.write( "7k+1 + 7·3(7k-1 - 3k-1) - 3k+1 = 40p\r\n" );
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document.write( "By (B) above we can replace the parentheses by 4q\r\n" );
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document.write( "7k+1 + 7·3(4q) - 3k+1 = 40p\r\n" );
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document.write( "7k+1 + 84q - 3k+1 = 40p\r\n" );
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document.write( "Rearrange the equation:\r\n" );
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document.write( "7k+1 - 3k+1 = 40p - 84q\r\n" );
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document.write( "7k+1 - 3k+1 = 4(10p - 21)q\r\n" );
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document.write( "The right side is divisible by 4, and therefore so is the left side.\r\n" );
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document.write( "QED\r\n" );
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document.write( "Edwin
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