document.write( "Question 1205219: 2) Determine the order of the following elements.
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Algebra.Com's Answer #841887 by ikleyn(52797) $\"\"$ $\"About$

You can put this solution on YOUR website!
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Soup of words.

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\n" ); document.write( "\n" ); document.write( "        A correct mathematical wording should be like THIS:\r
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\n" ); document.write( "\n" ); document.write( "(a)   In the multiplicative group   Z/(8Z)^*, find the order of element   7 (mod 8);\r
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\n" ); document.write( "\n" ); document.write( "(b)   In the multiplicative group   Z/(11Z)^*, find the order of element   5 (mod 11).
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document.write( "                     Solution\r\n" );
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document.write( "(a)  They want you find minimal positive integer n such that \r\n" );
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document.write( "           mod 8 = 1.\r\n" );
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document.write( "     Try  n= 1:   mod 8 = 7,  so, n= 1 does not work.\r\n" );
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document.write( "     Try  n= 2:   mod 8 = 49 mod 8 = 48 mod 8 + 1 mod 8 = 0 mod 8 + 1 mod 8 = 1 mod 8.  \r\n" );
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document.write( "     Thus,  n= 2 does work and is the ANSWER.\r\n" );
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document.write( "(b)  They want you find minimal positive integer n such that \r\n" );
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document.write( "           mod 11 = 1.\r\n" );
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document.write( "     Try  n= 1:   mod 11 = 5,  so, n= 1 does not work.\r\n" );
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document.write( "     Try  n= 2:   mod 11 = 25 mod 11 = 3 mod 11;   so, n= 2 does not work. \r\n" );
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document.write( "     Try  n= 3:   mod 11 = (3*5) mod 11 = 15 mod 11 = 4 mod 11;   so, n= 3 does not work. \r\n" );
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document.write( "     Try  n= 4:   mod 11 = (4*5) mod 11 = 20 mod 11 = 9 mod 11;   so, n= 4 does not work. \r\n" );
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document.write( "     Try  n= 5:   mod 11 = (9*5) mod 11 = 45 mod 11 = 1 mod 11;   so, n= 5 does work. \r\n" );
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document.write( "     Thus,  n= 5 does work and is the ANSWER.\r\n" );
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