document.write( "Question 803420: we have an unknown number of coin.If you make 77 strings of them, you are 50 coins short, but if you make 78 strings , it is exact.how many coins are there?
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Algebra.Com's Answer #484428 by Edwin McCravy(20055)\"\" \"About 
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document.write( "Let n be the answer.\r\n" );
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\n" ); document.write( "if you make 78 strings , it is exact.
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document.write( "So n is a multiple of 78, and therefore \r\n" );
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document.write( "n = 78p, for some natural number q\r\n" );
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\n" ); document.write( "If you make 77 strings of them, you are 50 coins short
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document.write( "So if 50 is added to n, the result is a multiple of 77\r\n" );
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document.write( "Therefore \r\n" );
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document.write( "   n + 50 = 77q for some integer\r\n" );
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document.write( "Substitute 78p for n\r\n" );
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document.write( "(1)   78p + 50 = 77q\r\n" );
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document.write( "The smaller coefficient of a letter in absolute value is 77\r\n" );
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document.write( "So we rewrite 78 and 50 in terms of their nearest multiples of 77\r\n" );
document.write( "We rewrite 78 as 77+1 and 50 as 77-27\r\n" );
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document.write( "  (77+1)p + 77-27 = 77q\r\n" );
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document.write( "77p + p + 77 - 27 = 77q\r\n" );
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document.write( "Divide through by 77\r\n" );
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document.write( "  p + \"p%2F77\" + 1 - \"27%2F77\" = q\r\n" );
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document.write( "Isolate the fractional terms\r\n" );
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document.write( "\"p%2F77\"-\"27%2F77\" = q - p - 1\r\n" );
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document.write( "The right side is an integer. Let that integer be A.\r\n" );
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document.write( "(2)     q - p - 1 = A\r\n" );
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document.write( "and so is the left side\r\n" );
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document.write( "            \"p%2F77\"-\"27%2F77\" = A  \r\n" );
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document.write( "Clear of fractions \r\n" );
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document.write( "            p - 27 = 77A\r\n" );
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document.write( "                 p = 77A + 27\r\n" );
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document.write( "Substitute in (2)\r\n" );
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document.write( "         q - p - 1 = A\r\n" );
document.write( "q - (77A + 27) - 1 = A\r\n" );
document.write( "  q - 77A - 27 - 1 = A\r\n" );
document.write( "      q - 77A - 28 = A\r\n" );
document.write( "                 q = 28 + 77A\r\n" );
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document.write( "So the solution to (1) is (p,q) = (77A+27,28+77A)\r\n" );
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document.write( "n = 78p = 78(77A+27) = 6006A+2106\r\n" );
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document.write( "The smallest solution is when A = 0, \r\n" );
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document.write( "n = 2106.\r\n" );
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document.write( "But there are infinitely many solutions, as A can be chosen as \r\n" );
document.write( "any positive integer.\r\n" );
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document.write( "Edwin
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