2005 AMC 10B Problem 13

Below is the professionally curated solution for Problem 13 of the 2005 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 10B solutions, or check the answer key.

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Concepts:multiple inclusion-exclusion counting integers in a range

Difficulty rating: 1370

13.

How many numbers between $1$ and $2005$ are integer multiples of $3$ or $4$ but not $12?$

$501$

$668$

$835$

$1002$

$1169$

Solution:

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 $(668 - 167) + (501 - 167) = 835$ numbers that are multiples of $3$ or $4$ but not $12.$

Thus, C is the correct answer.

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