2007 AIME I Problem 1

Question

How many positive perfect squares less than

10^6

are multiples of

24?

Answer

A perfect square is a multiple of

24 = 2^3 \cdot 3

exactly when its square root is a multiple of

12

Count the multiples of

12

whose squares are less than

10^6,

i.e. the multiples of

12

less than

1000

Since

24 = 2^3 \cdot 3,

a square

N^2

is a multiple of

24

exactly when

N

is a multiple of

12:

the factor

2^3

forces

N

to contain

2^2,

the factor

3

forces

N

to contain

3,

and conversely

(12m)^2 = 144m^2

is always a multiple of

24.

Also

N^2 \lt 10^6

exactly when

N \lt 1000.

The multiples of

12

less than

1000

are

12, 24, \ldots, 996,

and there are

\frac{996}{12} = 83

of them.

Problem 1 in other years