Difference between revisions of "1974 AHSME Problems/Problem 9"
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{{AHSME box|year=1974|num-b=8|num-a=10}} | {{AHSME box|year=1974|num-b=8|num-a=10}} | ||
| + | [[Category:Introductory Number Theory Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 11:42, 5 July 2013
Problem
The integers greater than one are arranged in five columns as follows:
![\[\begin{tabular}{c c c c c}\ & 2 & 3 & 4 & 5\\ 9 & 8 & 7 & 6 &\ \\ \ & 10 & 11 & 12 & 13\\ 17 & 16 & 15 & 14 &\ \\ \ & . & . & . & .\\ \end{tabular}\]](http://latex.artofproblemsolving.com/a/f/b/afbd2edf35ad29f633ac5cf8348f5cbe773fdc17.png)
(Four consecutive integers appear in each row; in the first, third and other odd numbered rows, the integers appear in the last four columns and increase from left to right; in the second, fourth and other even numbered rows, the integers appear in the first four columns and increase from right to left.)
In which column will the number 
fall?
Solution
We try pairing numbers with the column number they're in.
![\[\begin{tabular}{c c}\ \text{Number} & \text{Column Number} \\ 2 & 2\\ 3 & 3\\ 4 & 4\\ 5 & 5\\ 6 & 4\\ 7 & 3\\ 8 & 2\\ 9 & 1\\ 10 & 2\\ 11 & 3\\ 12 & 4\\ \end{tabular}\]](http://latex.artofproblemsolving.com/2/7/c/27c34e3fa32ba5b614d7dbc4bbeae41344bcac31.png)
Now we can see a pattern. The column number starts at 
, goes up to 
, then goes back down to 
, and repeats. Each of these blocks has a length of 
, so the column number has a period of 
. Since 
, 
is in the same column as , which is the second column. 
See Also
| 1974 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
