1974 AMC 12 Problem 9

Question

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

	$2$	$3$	$4$	$5$
$9$	$8$	$7$	$6$
	$10$	$11$	$12$	$13$
$17$	$16$	$15$	$14$
	$\vdots$	$\vdots$	$\vdots$	$\vdots$

(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?

first

second

third

fourth

fifth

Answer: B

Concepts:pattern recognition modular arithmetic

Difficulty rating: 1430

Small hint:

Look at where the displayed multiples of

8

occur

Big hint:

Each two-row cycle contains eight consecutive integers

Solution:

Each pair of rows is a repeating block of eight integers. In every such block, its multiple of

8

lies in the second column, as do

8

and

16.

Since

1000=125\cdot8,

it also lies in the second column.

Therefore, the correct answer is B.