2001 AMC 12 Problem 12

Question

How many positive integers not exceeding

2001

are multiples of

3

or

4

but not

5?

768

801

934

1067

1167

First count multiples of

3

or

4

using

\lfloor \frac{2001}{3} \rfloor

+ \lfloor \frac{2001}{4} \rfloor

- \lfloor \frac{2001}{12} \rfloor

Then subtract those also divisible by

5,

i.e. the multiples of

15

or

20

Multiples of

3

or

4

up to

2001

number

667 + 500 - 166 = 1001,

using

\lfloor \frac{2001}{3} \rfloor = 667,

\lfloor \frac{2001}{4} \rfloor = 500,

and

\lfloor \frac{2001}{12} \rfloor = 166.

Among these, the ones divisible by

5

are multiples of

15

or

20

:

133 + 100 - 33 = 200,

using

\lfloor \frac{2001}{15} \rfloor = 133,

\lfloor \frac{2001}{20} \rfloor = 100,

and

\lfloor \frac{2001}{60} \rfloor = 33.

The count is

1001 - 200 = 801.

Thus, the correct answer is B.

Problem 12 in other years