document.write( "Question 1183390: given set Z(26) = {0,1,2,3,...25}
\n" ); document.write( "Let a and b be integers. Consider the function f: Z(26) -> Z(26) defined by f(x) = ax + b (mod 26).
\n" ); document.write( "a)Prove that f(x) = 3x + b is bijective for any b.
\n" ); document.write( "b) find all the values of a for which the function f is bijective. \n" ); document.write( "

Algebra.Com's Answer #813693 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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document.write( "Z(26)  is the residue ring modulo 26.\r\n" );
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document.write( "    |    Since you use this denoting, I assume that you are familiar   |\r\n" );
document.write( "    |          with the term  \"the residue ring modulo 26\"             |\r\n" );
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document.write( "and with all other associated terms.\r\n" );
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document.write( "The map  f :  x ----> 3x + b  is the bijective for any \"b\", because 3 mod 26 is an invertible element of the ring Z(26).\r\n" );
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document.write( "Indeed, in this ring  3*9 = 1  (since 3*9 = 27 = 1 mod 26);  so, the element \"9\" of the ring Z(26) is inverse to the element \"3\".\r\n" );
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document.write( "It is the answer to question (a).\r\n" );
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document.write( "The answer to question (b) is    \"any number / (element of Z(26) ), which is mutually prime with /(or to) the number 26 \".\r\n" );
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document.write( "For example, the elements  \"1\", \"3\", \"5\", \"7\", \"9\", \"11\"  satisfy this criterion;  \r\n" );
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document.write( "the elements  \"2\", \"4\", \"6\", . . . , \"13\" do not satisfy the criterion.\r\n" );
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\n" ); document.write( "\n" ); document.write( "I intently presented the solution in terms of the ring theory, because your post was formulated in these terms.\r
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