document.write( "Question 1208625: A three digit number \"ABC\" is divided by the two digit number \"AC\". The quotient is 16 with no remainder. What is the largest possible number \"ABC\"? \n" ); document.write( "

Algebra.Com's Answer #847002 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "A three digit number \"ABC\" is divided by the two digit number \"AC\". The quotient is 16 with no remainder.
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document.write( "From the problem, we have \r\n" );
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document.write( "    100*A + 10*B + C = 16*(10A + C),   (1)\r\n" );
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document.write( "    1 <= A <= 9,   0 <= B, C <= 9.     (2)\r\n" );
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document.write( "Equation (1) is equivalent to\r\n" );
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document.write( "    100*A + 10*B + C = 160*A + 16C,\r\n" );
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document.write( "    60*A  - 10B + 15C = 0,\r\n" );
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document.write( "    12*A - 2B + 3C = 0.      (3)\r\n" );
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document.write( "From the last equation (3), we conclude that \r\n" );
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document.write( "    (a)  C is an even number 0, 2, 4, 6, or 8; \r\n" );
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document.write( "    (b)  B is a multiple of 3, so B is either 0, or, 3, or 6, or 9;\r\n" );
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document.write( "    (c)  2B - 3C >= 12.\r\n" );
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document.write( "From (c), if C= 0, then admittable values for B are 6 or 9;\r\n" );
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document.write( "          if C= 2, then admittable value for B is 9;\r\n" );
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document.write( "             C can not be 4, or 6, or 8  ( from inequalities (c) )\r\n" );
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document.write( "So, the only possible blocks of two digits BC can be 60, 90, 92.\r\n" );
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document.write( "If BC is 60, then from (3)  12*A - 2*6 + 3*0 = 0,  hence,  A = 1.  So, AC = 10, ABC = 160.\r\n" );
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document.write( "If BC is 90, then from (3)  12*A - 2*9 + 3*0 = 0,  hence, A = 18/12 = 1.5, which is not an integer number, so this case does not work.\r\n" );
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document.write( "If BC is 92, then from (3)  12*A - 2*9 + 3*2 = 0,  hence, A = 12/12 = 1.  So, AC = 12, ABC = 192.\r\n" );
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document.write( "Thus, the maximum possible number ABC is 192.    \r\n" );
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document.write( "ANSWER.  The only maximum possible number ABC under given condition is 192.\r\n" );
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