document.write( "Question 1207686: Let $x$ and $y$ be integers. If $x$ and $y$ satisfy $41x + 5y = 31$, then find the residue of $x$ modulo $5$. \n" ); document.write( "

Algebra.Com's Answer #845694 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Let x and y be integers. If x and y satisfy 41x + 5y = 31, then find the residue of x modulo 5.
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document.write( "You are given this equation\r\n" );
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document.write( "    41x + 5y = 31,    (1)\r\n" );
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document.write( "where x and y are integers.  Hence\r\n" );
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document.write( "    41x = 31 - 5y.    (2)\r\n" );
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document.write( "They want you find residue  {x mod 5} from modular equation (2).\r\n" );
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document.write( "It means that you consider equation (2)  as an equivalence by modulo 5.\r\n" );
document.write( "If so, you may forget about the term -5y, because it is 0 (zero) by the modulo 5.\r\n" );
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document.write( "So, your task is to find the residue {x mod 5} from the modular equation\r\n" );
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document.write( "    41x = 31 mod 5,\r\n" );
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document.write( "which is equivalent to \r\n" );
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document.write( "    41x = 1  mod 5.    (3)\r\n" );
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document.write( "The set of possible solutions is  x = {0 mod 5}, {1 mod 5}, {2 mod 5}, {3 mod 5} and {4 mod 5}.\r\n" );
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document.write( "    x = {0 mod 5} turns (3)  into  41*0 = 1 mod 5,  which is obviously wrong. \r\n" );
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document.write( "    x = {1 mod 5} turns (3)  into  41*1 = 1 mod 5,  which is obviously correct modular equation ( since 41 mod 5 = 1 mod 5). \r\n" );
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document.write( "    x = {2 mod 5} turns (3)  into  41*2 = 1 mod 5,  which is obviously wrong. \r\n" );
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document.write( "    x = {3 mod 5} turns (3)  into  41*3 = 1 mod 5,  which is obviously wrong. \r\n" );
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document.write( "    x = {4 mod 5} turns (3)  into  41*4 = 1 mod 5,  which is obviously wrong. \r\n" );
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document.write( "Thus of five possible solutions only one is valid:  x = {1 mod 5}.    ANSWER\r\n" );
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document.write( "So, in this problem, you can easily find the answer by the brute force method.\r\n" );
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\n" ); document.write( "\n" ); document.write( "The problem can be solved using more refined methods, but this reasoning is the number 1,
\n" ); document.write( "which you should know and understand as a basic method to start thinking.\r
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