2005 AMC 10B Problem 13

Question

How many numbers between

1

and

2005

are integer multiples of

3

or

4

but not

12?

501

668

835

1002

1169

Count multiples of

3

and of

4

separately, noting that multiples of

12

are the overlap

From each of the

3

-multiples and

4

-multiples, remove the

12

-multiples

Between

1

and

2005

there are

668

multiples of

3,

501

multiples of

4,

and

167

multiples of

12.

Every multiple of

12

is both a multiple of

3

and of

4,

so removing them from each group gives

\begin{aligned} &(668 - 167) \\ &\quad {}+ (501 - 167) = 835 \end{aligned}

numbers that are multiples of

3

or

4

but not

12.

Thus, C is the correct answer.

Problem 13 in other years