Algebra.Com's Answer #846195 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "When expanded as a decimal, the fraction  has a repetend (the repeating part of the decimal)
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document.write( "that begins right after the decimal point, and is 976 digits long.
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document.write( "If the last three digits of the repetend are ABC, compute the digits A, B, and C.
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document.write( "A=
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document.write( "B=
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document.write( "C=
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document.write( "Let = x. Since the repetend is 977 digits long, we see that  has the same decimal\r\n" );
document.write( "expansion as x. Hence, the number n =  is an integer number and it is (it represents) the \"repetend\".\r\n" );
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document.write( "But, from the other side, this number n is \r\n" );
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document.write( " 9999...999999\r\n" );
document.write( " n =  = --------------- .\r\n" );
document.write( " 977 \r\n" );
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document.write( " +---------------------------------------------------------------+\r\n" );
document.write( " | The numerator is the integer formed by 976 digits of \"9\". |\r\n" );
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document.write( "So, we can write 9999...999999 = 977*n, where n is some integer number. (1)\r\n" );
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document.write( "This number \"n\" is the \"repetend part\", and our goal is to find three last digits of n.\r\n" );
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document.write( " +-------------------------------------------------------------------+\r\n" );
document.write( " | From this writing (1), it is clear that the last digit of n is 7, |\r\n" );
document.write( " | providing 7*7 = 49 with the last digit 9; |\r\n" );
document.write( " | so, we can write n = 10m+7 for some integer m. |\r\n" );
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document.write( "Substitute it into (1). You will get\r\n" );
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document.write( " 9999...999999 = 977*(10m+7) = 977*10m + 6839.\r\n" );
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document.write( "Hence, 9999...999999 = 977*10m + 6839. (2)\r\n" );
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document.write( "Subtract 6839 from both sides of (2). You will get\r\n" );
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document.write( " 9999...9993160 = 977*10m (3)\r\n" );
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document.write( "In (3), divide both sides by 10. You will get\r\n" );
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document.write( " 9999...999316 = 977*m, (4)\r\n" );
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document.write( "where the number in the left side has the last 3 digits 316 and all other preceding digits are \"9\", and \"m\" is some integer.\r\n" );
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document.write( " +-------------------------------------------------------------------+\r\n" );
document.write( " | From this writing (4), it is clear that the last digit of m is 8, |\r\n" );
document.write( " | providing 7*8 = 56 with the last digit 6; |\r\n" );
document.write( " | so, we can write m = 10k+8 for some integer k. |\r\n" );
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document.write( "Substitute it into (4). You will get\r\n" );
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document.write( " 9999...999316 = 977*(10k+8) = 977*10k + 7816.\r\n" );
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document.write( "Hence, 9999...999316 = 977*10k + 7816. (5)\r\n" );
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document.write( "Subtract 7816 from both sides of (5). You will get\r\n" );
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document.write( " 9999...991500 = 977*10k (6)\r\n" );
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document.write( "In (6), divide both sides by 10. You will get\r\n" );
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document.write( " 9999...999150 = 977*k, (7)\r\n" );
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document.write( "where the number in the left side has the last 3 digits 134 and all other preceding digits are \"9\", and \"k\" is some integer.\r\n" );
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document.write( " +-------------------------------------------------------------------+\r\n" );
document.write( " | From this writing (7), it is clear that the last digit of k is 0, |\r\n" );
document.write( " | providing 7*0 = 0 with the last digit 0. |\r\n" );
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document.write( "Thus the last three digits in repetend part are 087.\r\n" );
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document.write( "ANSWER. The last three digits in repetend part are 087.\r\n" );
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document.write( " So, A = 0, B = 8, C = 7.\r\n" );
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document.write( "Solved.\r
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