document.write( "Question 898721: If one is the remainder when A to the second power is divided by 4, what would the remainder have to be if (A+5) to the second power is divided by 4? \n" ); document.write( "
Algebra.Com's Answer #555460 by Edwin McCravy(20081)\"\" \"About 
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If one is the remainder when A to the second power is divided by 4, what would
\n" ); document.write( "the remainder have to be if (A+5) to the second power is divided by 4?
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document.write( "Lemma 1:\r\n" );
document.write( "The square of every odd positive integer leaves remainder 1 when\r\n" );
document.write( "divided by 4:\r\n" );
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document.write( "Proof: Every odd integer can be expressed as 2n-1 for some positive integer n.\r\n" );
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document.write( "(2n-1)2= 4n2-4n+1=4(n2-n)+1 \r\n" );
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document.write( "That is 1 more than a multiple of 4 so it leaves remainder 1 when \r\n" );
document.write( "divided by 4.\r\n" );
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document.write( "Lemma 2:\r\n" );
document.write( "The square of every even positive integer leaves remainder 0 when\r\n" );
document.write( "divided by 4:\r\n" );
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document.write( "Proof: Every even integer can be expressed as 2n for some positive integer n.\r\n" );
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document.write( "(2n)2= 22n2 = 4n2  \r\n" );
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document.write( "That is a multiple of 4 so it leaves remainder 0 when \r\n" );
document.write( "divided by 4.\r\n" );
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document.write( "Therefore by those two lemmas, A can be any odd number (and cannot be an even\r\n" );
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document.write( "Therefore (A+5)2 = [(2n-1)+5]2 = (2n+4)2 = [2(n+2)]2.\r\n" );
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document.write( "That's the square of an even integer and by lemma 2 above it has remainder 0\r\n" );
document.write( "when divided by 4.\r\n" );
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document.write( "Answer: the remainder is 0\r\n" );
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document.write( "Edwin
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