document.write( "Question 1207684: Find all integers $n$, $0 \le n < 163$, such that $n$ is its own inverse modulo $8.$ \n" ); document.write( "

Algebra.Com's Answer #845696 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Find all integers n, 0 < n < 163, such that n is its own inverse modulo 8.
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document.write( "n is own inverse modulo 8 means\r\n" );
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document.write( "    n*n = 1 mod 8.\r\n" );
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document.write( "We consider numbers n modulo 8, so, it is enough to consider \r\n" );
document.write( "the congruence classes from  {1 mod 8}  to  {7 mod 8}.\r\n" );
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document.write( "n= 1:  n*n = 1*1 = 1 mod 8.       Hence,  n= 1 works.\r\n" );
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document.write( "n= 2:  n*n = 2*2 = 4 mod 8.       Hence,  n= 2 does not work.\r\n" );
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document.write( "n= 3:  n*n = 3*3 = 1 mod 8.       Hence,  n= 3  works.\r\n" );
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document.write( "n= 4:  n*n = 4*4 = 16 = 0 mod 8.  Hence,  n= 4 does not work.\r\n" );
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document.write( "n= 5:  n*n = 5*5 = 1 mod 8.       Hence,  n= 5  works.\r\n" );
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document.write( "n= 6:  n*n = 6*6 = 36 = 4 mod 8.  Hence,  n= 6  does not work.\r\n" );
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document.write( "n= 7:  n*n = 7*7 = 49 = 1 mod 8.  Hence,  n= 7  works.\r\n" );
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document.write( "ANSWER.  Among the congruence classes mod 8,  classes 1, 3, 5 and 7 work as own inverses mod 8.\r\n" );
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document.write( "         the rest of classes do not work as own inverses mod 8.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Do we need to explain the obvious fact that every mathematical problem requires using
\n" ); document.write( "the relevant units of measurement?\r
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\n" ); document.write( "\n" ); document.write( "In this problem we operate with equivalence classes (residues), these classes are from
\n" ); document.write( "{0 mod 8} to {7 mod 8}.\r
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\n" ); document.write( "\n" ); document.write( "Therefore, references to numbers greater than 8 are inappropriate and only show/demonstrate
\n" ); document.write( "the general mathematical illiteracy of the person who compiled this problem.\r
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