1974 AHSME Problems/Problem 9

Problem

The integers greater than one are arranged in five columns as follows:

$\begin{tabular}{c c c c c}\ & 2 & 3 & 4 & 5\\ 9 & 8 & 7 & 6 &\ \\ \ & 10 & 11 & 12 & 13\\ 17 & 16 & 15 & 14 &\ \\ \ & . & . & . & .\\ \end{tabular}$

(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 $1,000$ fall?

$\mathrm{(A)\ } \text{first} \qquad \mathrm{(B) \ }\text{second} \qquad \mathrm{(C) \ } \text{third} \qquad \mathrm{(D) \ } \text{fourth} \qquad \mathrm{(E) \ }\text{fifth}$

Solution

We try pairing numbers with the column number they're in.

$\begin{tabular}{c c}\ \text{Number} & \text{Column Number} \\ 2 & 2\\ 3 & 3\\ 4 & 4\\ 5 & 5\\ 6 & 4\\ 7 & 3\\ 8 & 2\\ 9 & 1\\ 10 & 2\\ 11 & 3\\ 12 & 4\\ \end{tabular}$

Now we can see a pattern. The column number starts at $2$ , goes up to $5$ , then goes back down to $1$ , and repeats. Each of these blocks has a length of $7$ , so the column number has a period of $8$ . Since $1000\equiv8(\bmod\,8)$ , $1000$ is in the same column as $8$ , which is the second column. $\boxed{\text{B}}$

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
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1974 AHSME Problems/Problem 9

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