document.write( "Question 1208759: Find the smallest integer which will divide over 45, 72, and 999 leaving remainder as 5, 2, and 9. \n" ); document.write( "

Algebra.Com's Answer #847200 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "Find the smallest integer which will divide over 45, 72, and 999 leaving remainder as 5, 2, and 9.
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\n" ); document.write( "\n" ); document.write( "Such an integer number does not exist.\r
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\n" ); document.write( "\n" ); document.write( "Below I will prove it mathematically.\r
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document.write( "Let assume that some integer x satisfies these congruences\r\n" );
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document.write( "    x = 5  (mod 45)     (1)\r\n" );
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document.write( "    x = 2  (mod 72)     (2)\r\n" );
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document.write( "    x = 9  (mod 999)    (3)\r\n" );
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document.write( "Notice that 999 = 9*111,  72 = 8*9.\r\n" );
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document.write( "Congruence (3) tells us that the number  x-9  is divisible by 999.\r\n" );
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document.write( "Hence,  x-9  is divisible by 9, too.\r\n" );
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document.write( "Next, congruence (2) tells us that the number  x-2  is divisible by 72.\r\n" );
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document.write( "Hence,  x-2  is divisible by 9, too.\r\n" );
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document.write( "It is just a contradiction, because the numbers  x-2  and  x-9  can not be divisible \r\n" );
document.write( "by 9 simultaneously - otherwise, their difference  (x-2) - (x-9) = 7 would be divisible by 9,\r\n" );
document.write( "which is not the case.\r\n" );
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document.write( "This contradiction proves  that the system of congruences (1), (2), (3) has no solutions in integer numbers.\r\n" );
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document.write( "In this proof, I used congruences (2) and (3).\r\n" );
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document.write( "But for proving this statement, I could equally use any two congruences of these three given congruences.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved, with the proof and explanations.\r
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\n" ); document.write( "\n" ); document.write( "So, this problem is a trap, in some sense.\r
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\n" ); document.write( "\n" ); document.write( "         The meaning of this problem is to teach a reader\r
\n" ); document.write( "\n" ); document.write( "                    (a) to recognize such traps \r
\n" ); document.write( "\n" ); document.write( "             and (b) to prove that the problem is a trap.\r
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