Topic : Counting-Vendiagram
from: category_eng
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2. '

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

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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 leftlfloordfrac{2005}{3}<br />
ight<br />
floor+leftlfloordfrac{2005}{4}<br />
ight<br />
floor-2*leftlfloordfrac{2005}{12}<br />
ight<br />
floor... (where lfloor x <br />
floor denotes the floor function).