2000 AMC 12 Problems/Problem 16
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Problem
A checkerboard of 
rows and 
columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered 
, the second row 
, and so on down the board. If the board is renumbered so that the left column, top to bottom, is 
, the second column 
and so on across the board, some squares have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system).
Solution
Let 
denote the square in row 
and column 
. Under the first ordering this square would have a value of 
. Under the second ordering this square would have a value of 
. Equating, 
. The pairs that fit this equation are 
; their corresponding values sum up to 
.
See also
| 2000 AMC 12 (Problems • Resources) | ||
| Preceded by Problem 15 | Followed by Problem 17 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
