document.write( "Question 1207719: Let $m$ and $n$ be non-negative integers. If $m = 6n + 2$, then what integer between $0$ and $m$ is the inverse of $2$ modulo $m$? Answer in terms of $n$. \n" ); document.write( "

Algebra.Com's Answer #845759 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Let m and n be non-negative integers. If m = 6n + 2, then what integer between 0 and m
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document.write( "Notice that m = 6n+2 is an even integer number, for any integer number n.\r\n" );
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document.write( "Let's assume that an integer x between 0 and m is the inverse of 2 modulo m.\r\n" );
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document.write( "It means that 2*x = 1 mod m, which is the same as to say that\r\n" );
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document.write( "    2x - 1 is a multiple of m :   2x - 1 = k*m.\r\n" );
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document.write( "But 2x is an even number, and k*m is an even number, since \"m\" is even.\r\n" );
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document.write( "Therefore, this equality  2x - 1 = km  is not possible with integer x and \"k\".\r\n" );
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document.write( "Hence, there is NO any integer between 0 and m which is inverse of 2 modulo m.\r\n" );
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document.write( "ANSWER.  There is NO any integer between 0 and m which is inverse of 2 modulo m.\r\n" );
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\n" ); document.write( "\n" ); document.write( "What I proved in my post, in terms of abstract algebra is THIS general statement:\r
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\n" ); document.write( "\n" ); document.write( " In the ring of integers modulo m, Z/m, where \" m \" is an even number,\r
\n" ); document.write( "\n" ); document.write( " the class {2 mod m} is NOT an invertible element.\r
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\n" ); document.write( "\n" ); document.write( "In opposite, in such a ring, the class {2 mod m} is a divisor of zero;
\n" ); document.write( "and it is well known fact of abstract algebra that in a ring a divisor of zero can not be invertible.\r
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