document.write( "Question 459304: Find the smallest possible integer which, when divided by 3 has a remainder of 1, when divided by 4 has a remainder of 2, and when divided by 5 has a remainder of 3. \n" ); document.write( "
Algebra.Com's Answer #314985 by Edwin McCravy(20054) $\"\"$ $\"About$
You can put this solution on YOUR website!
\r\n" );
document.write( "Here it is worked out using algebra:\r\n" );
document.write( "\r\n" );
document.write( "Let N be the answer.  Then there exist positive integers A, B, and C\r\n" );
document.write( "such that:\r\n" );
document.write( "\r\n" );
document.write( "N = 3A + 1 = 4B + 2 = 5C + 3\r\n" );
document.write( "\r\n" );
document.write( "3A + 1 = 4B + 2\r\n" );
document.write( "\r\n" );
document.write( "3A - 4B = 1\r\n" );
document.write( "\r\n" );
document.write( "Principle  P:\r\n" );
document.write( "This is the principle of\r\n" );
document.write( "\r\n" );
document.write( "1. Writing the absolute value  of \r\n" );
document.write( "each integer or coefficient in an equation in terms \r\n" );
document.write( "of its nearest multiple of the coefficient in absolute \r\n" );
document.write( "value that occurs in the equation,considering 0 as a \r\n" );
document.write( "multiple of every integer,\r\n" );
document.write( "2. Removing parentheses\r\n" );
document.write( "3. Dividing every term by that least coefficient in\r\n" );
document.write( "absolute value.\r\n" );
document.write( "4. Isolating all fractional \r\n" );
document.write( "terms.\r\n" );
document.write( "5. Setting both sides equal to a new positive integer \r\n" );
document.write( "variable:\r\n" );
document.write( "\r\n" );
document.write( "Using principle P on\r\n" );
document.write( "\r\n" );
document.write( "3A - 4B = 1\r\n" );
document.write( "\r\n" );
document.write( "3A - (3 + 1)B = 1\r\n" );
document.write( "\r\n" );
document.write( "3A - 3B - B = 1\r\n" );
document.write( "\r\n" );
document.write( "A - B - B/3 = 1/3\r\n" );
document.write( "\r\n" );
document.write( "A - B = 1/3 + B/3\r\n" );
document.write( "\r\n" );
document.write( "Let D = A - B = 1/3 + B/3\r\n" );
document.write( "\r\n" );
document.write( "D = 1/3 + B/3\r\n" );
document.write( "\r\n" );
document.write( "3D = 1 + B\r\n" );
document.write( "\r\n" );
document.write( "(1)    B = 3D - 1\r\n" );
document.write( "\r\n" );
document.write( "D = A - B \r\n" );
document.write( "D = A - (3D - 1)\r\n" );
document.write( "D = A - 3D + 1\r\n" );
document.write( "4D - 1 = A\r\n" );
document.write( "\r\n" );
document.write( "(2)   A = 4D - 1\r\n" );
document.write( "\r\n" );
document.write( "       4B + 2 = 5C + 3\r\n" );
document.write( "4(3D - 1) + 2 = 5C + 3\r\n" );
document.write( "  12D - 4 + 2 = 5C + 3\r\n" );
document.write( "      12D - 2 = 5C + 3\r\n" );
document.write( "     12D - 5C = 5\r\n" );
document.write( "\r\n" );
document.write( "Use principle P on that equation:\r\n" );
document.write( "\r\n" );
document.write( " (10+2)D - 5C = 5\r\n" );
document.write( "10D + 2D - 5C = 5\r\n" );
document.write( "2D + 2D/5 - C = 1\r\n" );
document.write( "2D/5 = C - 2D + 1\r\n" );
document.write( "\r\n" );
document.write( "2D/5 = C - 2D + 1 = E\r\n" );
document.write( "\r\n" );
document.write( "C - 2D + 1 = E\r\n" );
document.write( "\r\n" );
document.write( "(3)   C = E + 2D - 1\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2D/5 = E\r\n" );
document.write( "2D = 5E\r\n" );
document.write( "\r\n" );
document.write( "Use the principle P again:\r\n" );
document.write( "\r\n" );
document.write( "2D = (4+1)E\r\n" );
document.write( "2D = 4E + E\r\n" );
document.write( "D = 2E + E/2\r\n" );
document.write( "D - 2E = E/2\r\n" );
document.write( "\r\n" );
document.write( "E/2 = D - 2E = F\r\n" );
document.write( "\r\n" );
document.write( "E/2 = F,  D - 2E = F\r\n" );
document.write( "\r\n" );
document.write( "E = 2F\r\n" );
document.write( "\r\n" );
document.write( "D - 2(2F) = F\r\n" );
document.write( "D - 4F = F\r\n" );
document.write( "(4) D = 5F\r\n" );
document.write( "\r\n" );
document.write( "From (3) \r\n" );
document.write( "\r\n" );
document.write( "C = E + 2D - 1\r\n" );
document.write( "C = 2F + 2(5F) - 1\r\n" );
document.write( "C = 2F + 10F - 1\r\n" );
document.write( "C = 12F - 1\r\n" );
document.write( "\r\n" );
document.write( "(5)  C = 12F - 1\r\n" );
document.write( "\r\n" );
document.write( "From (1)    \r\n" );
document.write( "\r\n" );
document.write( "B = 3D - 1\r\n" );
document.write( "B = 3(5F) - 1\r\n" );
document.write( "(6)  B = 15F - 1\r\n" );
document.write( "\r\n" );
document.write( "D = A - B\r\n" );
document.write( "5F = A - (15F - 1)\r\n" );
document.write( "5F = A - 15F + 1\r\n" );
document.write( "\r\n" );
document.write( "20F - 1 = A\r\n" );
document.write( "\r\n" );
document.write( "A = 20F - 1\r\n" );
document.write( "B = 15F - 1\r\n" );
document.write( "C = 12F - 1\r\n" );
document.write( "\r\n" );
document.write( "Therefore the smallest values of A, B, and C\r\n" );
document.write( "will be when F=11\r\n" );
document.write( "\r\n" );
document.write( "A = 20F - 1 = 20 - 1 = 19\r\n" );
document.write( "B = 15F - 1 = 15 - 1 = 14\r\n" );
document.write( "C = 7F - 1 = 12 - 1 = 11\r\n" );
document.write( "\r\n" );
document.write( "N = 3A + 1 = 4B + 2 = 5C + 3\r\n" );
document.write( "\r\n" );
document.write( "N = 3(19) + 1 = 4(14) + 2 = 5(11) + 3\r\n" );
document.write( "\r\n" );
document.write( "N = 58 = 58 = 58\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( " \n" ); document.write( "
\n" );