\r\n" );
document.write( "\r\n" );
document.write( "5n + 10d + 25q = 100\r\n" );
document.write( "\r\n" );
document.write( "Divide through by 5\r\n" );
document.write( "\r\n" );
document.write( " n + 2d + 5q = 20\r\n" );
document.write( "\r\n" );
document.write( " n = 20 - 2d - 5q\r\n" );
document.write( "\r\n" );
document.write( " 0 < n < 20\r\n" );
document.write( " \r\n" );
document.write( " 0 < 20 - 2d - 5q < 20\r\n" );
document.write( "\r\n" );
document.write( " -20 < -2d - 5q < 0\r\n" );
document.write( "\r\n" );
document.write( " 20 > 2d + 5q > 0\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "That has the graph of all the lattice points (d,q) (where d and\r\n" );
document.write( "q are non-negative integers on or inside the triangle whose\r\n" );
document.write( "corners (d,q) are (0,0), (10,0) and (0,4)\r\n" );
document.write( "\r\n" );
document.write( "The graph below has x as d and y as q.\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Now we find how many lattice points (d,q) are on the line\r\n" );
document.write( "\r\n" );
document.write( "2d + 5q = 20\r\n" );
document.write( "\r\n" );
document.write( "Write 5q as 4q + q\r\n" );
document.write( "\r\n" );
document.write( "2d + 4q + q = 20\r\n" );
document.write( "\r\n" );
document.write( "Divide through by 2\r\n" );
document.write( "\r\n" );
document.write( "d + 2q + q/2 = 10\r\n" );
document.write( "\r\n" );
document.write( "q/2 = 10 - d - 2q\r\n" );
document.write( "\r\n" );
document.write( "The right side is an integer, and so is the left side.\r\n" );
document.write( "The left side is non-negative so both sides are \r\n" );
document.write( "non-negative integers, say A,\r\n" );
document.write( "\r\n" );
document.write( "q/2 = A \r\n" );
document.write( "\r\n" );
document.write( "q = 2A\r\n" );
document.write( "\r\n" );
document.write( "10 - d - 2q = A\r\n" );
document.write( "10 - d - 2(2A) = A\r\n" );
document.write( "10 - d - 4A = A\r\n" );
document.write( "10 - 5A = d\r\n" );
document.write( "\r\n" );
document.write( "0 < d < 10\r\n" );
document.write( "0 < 10-5A < 10\r\n" );
document.write( "0 < 2 - A < 2\r\n" );
document.write( "-2 < -A < 0\r\n" );
document.write( " 2 > A > 0\r\n" );
document.write( "\r\n" );
document.write( "Thus A = 0,1,2\r\n" );
document.write( "\r\n" );
document.write( "That means that there are 3 lattice points (d,q) on the line\r\n" );
document.write( "\r\n" );
document.write( "d = 10-5A, q = 2A \r\n" );
document.write( "\r\n" );
document.write( "(10,0), (5,2), (0,4)\r\n" );
document.write( "\r\n" );
document.write( "The number of lattice points in or on the rectangle with corners\r\n" );
document.write( "\r\n" );
document.write( "(0,0), (10,0), (10,4), (0,4)\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "is 11x5 = 55 (that is, there are 11 integers from 0 through 10 inclusive,\r\n" );
document.write( "and 5 integers from 0 through 4 inclusive.\r\n" );
document.write( "\r\n" );
document.write( "There are 3 lattice points on the diagonal. So there are 55-3= 52\r\n" );
document.write( "lattice points in or on the rectangle that are not on the diagonal,\r\n" );
document.write( "and since there are as many lattice points above the diagonal as \r\n" );
document.write( "there are below the line, there are 26 latice points below the line. \r\n" );
document.write( "Adding the 3 lattice points on the diagonal gives us 29 lattice points.\r\n" );
document.write( "\r\n" );
document.write( "Thus there are 29 solutions.\r\n" );
document.write( "\r\n" );
document.write( " 1. 0 quarters, 0 dimes, and 20 nickels.\r\n" );
document.write( " 2. 0 quarters, 1 dimes, and 18 nickels.\r\n" );
document.write( " 3. 0 quarters, 2 dimes, and 16 nickels.\r\n" );
document.write( " 4. 0 quarters, 3 dimes, and 14 nickels.\r\n" );
document.write( " 5. 0 quarters, 4 dimes, and 12 nickels.\r\n" );
document.write( " 6. 0 quarters, 5 dimes, and 10 nickels.\r\n" );
document.write( " 7. 0 quarters, 6 dimes, and 8 nickels.\r\n" );
document.write( " 8. 0 quarters, 7 dimes, and 6 nickels.\r\n" );
document.write( " 9. 0 quarters, 8 dimes, and 4 nickels.\r\n" );
document.write( "10. 0 quarters, 9 dimes, and 2 nickels.\r\n" );
document.write( "11. 0 quarters, 10 dimes, and 0 nickels.\r\n" );
document.write( "12. 1 quarters, 0 dimes, and 15 nickels.\r\n" );
document.write( "13. 1 quarters, 1 dimes, and 13 nickels.\r\n" );
document.write( "14. 1 quarters, 2 dimes, and 11 nickels.\r\n" );
document.write( "15. 1 quarters, 3 dimes, and 9 nickels.\r\n" );
document.write( "16. 1 quarters, 4 dimes, and 7 nickels.\r\n" );
document.write( "17. 1 quarters, 5 dimes, and 5 nickels.\r\n" );
document.write( "18. 1 quarters, 6 dimes, and 3 nickels.\r\n" );
document.write( "19. 1 quarters, 7 dimes, and 1 nickels.\r\n" );
document.write( "20. 2 quarters, 0 dimes, and 10 nickels.\r\n" );
document.write( "21. 2 quarters, 1 dimes, and 8 nickels.\r\n" );
document.write( "22. 2 quarters, 2 dimes, and 6 nickels.\r\n" );
document.write( "23. 2 quarters, 3 dimes, and 4 nickels.\r\n" );
document.write( "24. 2 quarters, 4 dimes, and 2 nickels.\r\n" );
document.write( "25. 2 quarters, 5 dimes, and 0 nickels.\r\n" );
document.write( "26. 3 quarters, 0 dimes, and 5 nickels.\r\n" );
document.write( "27. 3 quarters, 1 dimes, and 3 nickels.\r\n" );
document.write( "28. 3 quarters, 2 dimes, and 1 nickels.\r\n" );
document.write( "29. 4 quarters, 0 dimes, and 0 nickels.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
