1962 AMC 12 Problem 38

Question

The population of Nosuch Junction at one time was a perfect square. Later, with an increase of

100,

the population was one more than a perfect square. Now, with an additional increase of

100,

the population is again a perfect square.

The original population is a multiple of:

3

7

9

11

17

Write the populations as

a^2,

b^2+1,

and

c^2

From

b^2-a^2=99,

test the positive factor pairs of

99

, then impose

c^2-a^2=200

Let the original population be

a^2.

Then

\begin{aligned} b^2-a^2&=99,\\ c^2-a^2&=200. \end{aligned}

From

(b-a)(b+a)=99,

the positive possibilities for

a

are

49,

15,

and

1.

Checking the second condition, only

a=49

works, since

49^2+200=2601=51^2.

The population is

49^2=7^4,

a multiple of

7.

Thus, the correct answer is B.

Problem 38 in other years