2001 AMC 10 Problem 25

Question

How many positive integers not exceeding

2001

are multiples of

3

or

4

but not

5?

768

801

934

1067

1167

Count the multiples of

3

or

4,

then remove those that are also multiples of

5

Use inclusion-exclusion with multiples of

3,

4,

12

and then

15,

20,

60

Multiples of

3

or

4

up to

2001:

\left\lfloor\tfrac{2001}{3}\right\rfloor

+\left\lfloor\tfrac{2001}{4}\right\rfloor

-\left\lfloor\tfrac{2001}{12}\right\rfloor

=667+500-166

=1001.

Among these, remove the multiples of

5:

there are

133

multiples of

15

and

100

multiples of

20,

re-adding the

33

multiples of

60:

133+100-33=200.

So the count is

1001-200=801.

Thus, the correct answer is B.

Problem 25 in other years