2005 AMC 10B Problems/Problem 13

Problem

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

$\mathrm{(A)} 501 \qquad \mathrm{(B)} 668 \qquad \mathrm{(C)} 835 \qquad \mathrm{(D)} 1002 \qquad \mathrm{(E)} 1169$

Solution

We can use the Principle of Inclusion-Exclusion to solve the problem as follows: We can count the number of multiples of $3$ that are less than $2005$, add the number of multiples of $4$ that are less than $2005$, and subtract the number of multiples of $12$ twice that are less than $2005$ (since those are counted twice in each of the $3$ and $4$ cases). Calculating, we get $\left\lfloor\dfrac{2005}{3}\right\rfloor+\left\lfloor\dfrac{2005}{4}\right\rfloor-2*\left\lfloor\dfrac{2005}{12}\right\rfloor=668+501-334=\boxed{\mathrm{(c)}\ 835}$ (where $\lfloor x \rfloor$ denotes the floor function).

See Also

2005 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
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All AMC 10 Problems and Solutions

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