document.write( "Question 935274: This is a Number Theory proof:\r
\n" ); document.write( "\n" ); document.write( "If N = abc + 1, prove that (N, a) = (N, b) = (N, c) = 1.\r
\n" ); document.write( "\n" ); document.write( "(N, a) means \"the greatest common divisor of N and a.\"\r
\n" ); document.write( "\n" ); document.write( "I have started the proof like this:\r
\n" ); document.write( "\n" ); document.write( "Let d = (N, a).
\n" ); document.write( "Since d = (N, a), then d | N and d | a. \"d divides N\" and \"d divides a\".\r
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Algebra.Com's Answer #568511 by KMST(5328) $\"\"$ $\"About$

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Then $\"a%2Fd=x\"$ and $\"N%2Fd=M\"$ are integers.
\n" ); document.write( " $\"N%2Fd=M\"$ ---> $\"N=dM\"$
\n" ); document.write( " $\"a%2Fd=x\"$ ---> $\"a=dx\"$
\n" ); document.write( " $\"system%28a=dx%2CN=abc%2B1%29\"$ ---> $\"N=dxbc%2B1\"$
\n" ); document.write( " $\"system%28N=dM%2CN=dxbc%2B1%29\"$ ---> $\"dM=dxbc%2B1\"$ ---> $\"dM-dxbc=1\"$ ---> $\"d%28M-xbc%29=1\"$
\n" ); document.write( "Since $\"M\"$ , $\"x\"$ , $\"b\"$ , and $\"c\"$ are all integers, so is $\"M-xbc\"$ ,
\n" ); document.write( "and since the product of integers $\"d\"$ and $\"M-xbc\"$ is $\"1\"$ ,
\n" ); document.write( "they must both be $\"1\"$ :
\n" ); document.write( " $\"M-xbc=1\"$ and more importantly $\"d=1\"$ .
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\n" ); document.write( "The way that is proven for $\"a\"$ ,
\n" ); document.write( "it can be proven for $\"b\"$ and $\"c\"$
\n" ); document.write( "(but in a raesonable world it should not be required),
\n" ); document.write( "because they all play the same role with different names. \n" ); document.write( "

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