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\n" ); document.write( "Question:
\n" ); document.write( "Show that n^2 -1 is divisible by 8, if n is an odd positive integer.
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\n" ); document.write( "Solution:
\n" ); document.write( "If n is odd and positive, we define
\n" ); document.write( "n=2k+1 where k is a non-negative integer.
\n" ); document.write( "from which we substitute, expand and factor:
\n" ); document.write( "n²-1
\n" ); document.write( "=(2k+1)²-1
\n" ); document.write( "=4k²+4k+1-1
\n" ); document.write( "=4k²+4k
\n" ); document.write( "=4k(k+1)
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\n" ); document.write( "Since k is a non-negative integer, we have two possible cases:
\n" ); document.write( "1. k is odd, in which case (k+1) is even, and equal to 2q (q=non-negative integer)
\n" ); document.write( "=> n²-1=4k(2q)=8kq (where both k and q are non-negative integers)
\n" ); document.write( " therefore 8 divides n²-1
\n" ); document.write( "2. k is even, then k=2q (q=non-negative integer)
\n" ); document.write( "=> n²-1=4(2q)(k+1)=8q(k+1) (where both k and q are non-negative integers)
\n" ); document.write( " therefore 8 divides n²-1
\n" ); document.write( "Since in both cases, 8 divides n²-1, therefore it is proved that 8 divides n²-1 in for all positive values of n.
\n" ); document.write( " \n" ); document.write( "