document.write( "Question 1188394: In the sequence 457, 473, 487, 506, ... each term is equal to the previous term
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Algebra.Com's Answer #819474 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "In the sequence 457, 473, 487, 506, ... each term is equal to the previous term
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document.write( "First term, 457, has the sum of digits 4+5+7 = 16 = 1 (mod 3).\r\n" );
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document.write( "So, according to the divisibility rule by 3, the number 457 itself is 1 (mod 3).\r\n" );
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document.write( "    Therefore, the second term is 1+1 = 2 (mod 3).\r\n" );
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document.write( "    It means that the sum of the digits of the second term is 2 (mod 3).\r\n" );
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document.write( "It implies that the third term is 2+2 = 4 = 1 (mod 3)\r\n" );
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document.write( "It implies, in turn, that the fourth term is 2 (mod 3).\r\n" );
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document.write( "Thus we see that the given sequence of numbers is CYCLIC 1, 2, 1, 2 by modulo 3,\r\n" );
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document.write( "and it is true NOT ONLY for the first four terms, but IT IS TRUE for all other consecutive terms.\r\n" );
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document.write( "Now look at the given optional numbers.\r\n" );
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document.write( "Using the rule of divisibility by 3, you can easily check that the ONLY NUMBER of them \r\n" );
document.write( "which is DIVISIBLE by 3 is the number 4569.\r\n" );
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document.write( "So, we can conclude that this number, 4569, DOES NOT BELONG to the given sequence.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Edwin, this sequence follows the rule\r
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document.write( "                 = 1 (mod 3)\r\n" );
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document.write( "     = 1 + 1     = 2 (mod 3)\r\n" );
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document.write( "     = 2 + 2 = 4 = 1 (mod 3)\r\n" );
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document.write( "     = 1 + 1     = 2 (nod 3)\r\n" );
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\n" ); document.write( "\n" ); document.write( "and so on, and so on CYCLICALLY . . . , making the statement OBVIOUS.\r
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