document.write( "Question 411738: Show that any prime number greater than 3 has remainder 1 or 5 when divided by 6; \n" ); document.write( "
Algebra.Com's Answer #289358 by Edwin McCravy(20060)\"\" \"About 
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document.write( "Show that any prime number greater than 3 has remainder 1 or 5 \r\n" );
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document.write( "Every prime number greater than 3 is odd and not divisible by 3.\r\n" );
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document.write( "We will prove a stronger theorem:\r\n" );
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document.write( "Stronger theorem:\r\n" );
document.write( "Any odd integer greater than 3 which is not divisible by 3 \r\n" );
document.write( "has remainder of 1 or 5 when divided by 6\r\n" );
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document.write( "Every integer greater than 3 which is not divisible by 3 is \r\n" );
document.write( "of the form 3n+1 or 3n+2.\r\n" );
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document.write( "If n is even, say n = 2k then 3n+1 = 3(2k)+1 = 6k+1, which leaves\r\n" );
document.write( "a remainder of 1 when divided by 6\r\n" );
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document.write( "If n is even, say n = 2k then 3n+2 = 3(2k)+2 = 6k+2, which is even,\r\n" );
document.write( "so we don't need to consider this case.\r\n" );
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document.write( "If n is odd, say n = 2k+1, then 3n+1 = 3(2k+1)+1 = 6k+3+1 = 6k+4,\r\n" );
document.write( "which is even, so we don't need to consider this case either.\r\n" );
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document.write( "If n is odd, say n = 2k+1, then 3n+2 = 3(2k+1)+2 = 6k+3+2 = 6k+5, \r\n" );
document.write( "which leaves a remainder of 5.\r\n" );
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document.write( "So the theorem is proved.\r\n" );
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document.write( "Edwin
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