1959 AMC 12 Problem 14

Question

Given the set

S

whose elements are zero and the even integers, positive and negative. Of the five operations applied to any pair of elements:

(1)

addition,

(2)

subtraction,

(3)

multiplication,

(4)

division,

(5)

finding the arithmetic mean (average), those operations that yield only elements of

S

are:

all

1,

2,

3,

4

1,

2,

3,

5

1,

2,

3

1,

3,

5

Test whether each operation always takes two even integers to an even integer

Use small counterexamples for division and averaging

Even integers are closed under addition, subtraction, and multiplication. Division fails, since

\frac{2}{4}=\frac{1}{2}\notin S.

Averaging fails, since the mean of

0

and

2

is

1\notin S.

Thus only operations

1,

2,

and

3

always work, and the correct answer is D.

Problem 14 in other years