document.write( "Question 1041341: What least number, when divided by 26,36 and 56, leaves the remainders 6,16,36 respectively ? \n" ); document.write( "
Algebra.Com's Answer #656333 by Edwin McCravy(20054) $\"\"$ $\"About$
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document.write( "n = 26a+6 = 36b+16 = 56c+36\r\n" );
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document.write( "We solve the Diophantine equation\r\n" );
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document.write( "36b+16 = 56c+36\r\n" );
document.write( "  9b+4 = 14c+9\r\n" );
document.write( "    9b = 14c+5\r\n" );
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document.write( "9 is the least integer in that equation\r\n" );
document.write( "in absolute value, so write all the other\r\n" );
document.write( "integers in terms of a near multiple\r\n" );
document.write( "of 9.  So we write 14 as 18-4, and 5 is\r\n" );
document.write( "near 9 so we just leave it as it is.\r\n" );
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document.write( "9b = (18-4)c+5\r\n" );
document.write( "9b = 18c-4c+5\r\n" );
document.write( "Divide through by 9\r\n" );
document.write( "b = 2c-4c/9+5/9\r\n" );
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document.write( "Get the fraction on the left and other terms\r\n" );
document.write( "on the right:\r\n" );
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document.write( "4c/9-5/9 = 2c-b\r\n" );
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document.write( "The right side is an integer, say P, so\r\n" );
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document.write( "4c/9-5/9 = 2c-b = P\r\n" );
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document.write( "4c/9-5/9 = P\r\n" );
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document.write( "4c-5 = 9P\r\n" );
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document.write( "4 is the least integer in that equation\r\n" );
document.write( "in absolute value, so write all the other\r\n" );
document.write( "integers in terms of their nearest multiple\r\n" );
document.write( "of 4.  So we write 5 as 4+1, write 9 as 8+1.\r\n" );
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document.write( "4c-(4+1) = (8+1)P\r\n" );
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document.write( "4c-4-1 = 8P+P\r\n" );
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document.write( "Divide through by 4\r\n" );
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document.write( "c-1-1/4 = 2P+P/4\r\n" );
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document.write( "Get the fraction on the right, other terms\r\n" );
document.write( "on the left:\r\n" );
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document.write( "c-1-2P = P/4+1/4\r\n" );
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document.write( "The left side is an integer, say Q, so\r\n" );
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document.write( "c+1-2P = P/4+1/4 = Q\r\n" );
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document.write( "P/4+1/4 = Q\r\n" );
document.write( "    P+1 = 4Q\r\n" );
document.write( "      P = 4Q-1\r\n" );
document.write( "c-1-2P = Q\r\n" );
document.write( "c-1-2(4Q-1) = Q\r\n" );
document.write( "c-1-8Q+2 = Q\r\n" );
document.write( "  c+1-8Q = Q\r\n" );
document.write( "       c = 9Q-1\r\n" );
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document.write( "9b = 14c+5\r\n" );
document.write( "9b = 14c+5\r\n" );
document.write( "9b = 14(9Q-1)+5\r\n" );
document.write( "9b = 126Q-14+5\r\n" );
document.write( "9b = 126Q-9\r\n" );
document.write( " b = 14Q-1\r\n" );
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document.write( "26a+6 = 36b+16\r\n" );
document.write( "26a = 36b+10\r\n" );
document.write( "13a = 18b+5\r\n" );
document.write( "13a = 18(14Q-1)+5\r\n" );
document.write( "13a = 252Q-18+5\r\n" );
document.write( "13a = 252Q-13\r\n" );
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document.write( "13 is the least integer in that equation\r\n" );
document.write( "in absolute value, so write all the other\r\n" );
document.write( "integers in terms of their nearest multiple\r\n" );
document.write( "of 13.  So we write 252 as 247+5.\r\n" );
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document.write( "13a = (247+5)Q-13\r\n" );
document.write( "13a = 247Q+5Q-13\r\n" );
document.write( "Divide through by 13\r\n" );
document.write( "  a = 19Q+5Q/13-1\r\n" );
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document.write( "Get the fraction on the right and other\r\n" );
document.write( "terms left:\r\n" );
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document.write( "a+1-19Q = 5Q/13\r\n" );
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document.write( "The left side is an integer, say R, so\r\n" );
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document.write( "a+1-19Q = 5Q/13 = R\r\n" );
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document.write( "5Q/13 = R\r\n" );
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document.write( "5Q = 13R\r\n" );
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document.write( "We could continue as before but this equation is\r\n" );
document.write( "simpler and we see that the smallest integers Q \r\n" );
document.write( "and R that solve that are Q=13 and R=5.\r\n" );
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document.write( "So a+1-19Q = R\r\n" );
document.write( "   a+1-19(13) = 5\r\n" );
document.write( "   a+1-247 = 5\r\n" );
document.write( "    a-246 = 5\r\n" );
document.write( "        a = 251\r\n" );
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document.write( "And since n = 26a+6\r\n" );
document.write( "          n = 26(251)+6\r\n" );
document.write( "          n = 6526+6\r\n" );
document.write( "          n = 6532\r\n" );
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document.write( "Checking:\r\n" );
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document.write( "    251          181          116\r\n" );
document.write( "26)6532      36)6532      56)6532\r\n" );
document.write( "   52           36           56\r\n" );
document.write( "   133          293           93\r\n" );
document.write( "   130          288           56 \r\n" );
document.write( "     32           52          372\r\n" );
document.write( "     26           36          336\r\n" );
document.write( "      6           16           36  <-remainders check!\r\n" );
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document.write( "Edwin
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