2005 AMC 10A Problem 22

Problem 22 of 25HarderNumber Theory Counting & Probability

Let

S

be the set of the

2005

smallest positive multiples of

4,

and let

T

be the set of the

2005

smallest positive multiples of

6.

How many elements are common to

S

and

T?

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Solution

The elements common to

S

and

T

are the multiples of

\operatorname{lcm}(4,6) = 12.

Now

S

contains multiples of

4

up to

4 \cdot 2005 = 8020,

while

T

reaches up to

6 \cdot 2005 = 12{,}030,

so the common elements are the multiples of

12

not exceeding

8020.

There are

\left\lfloor \dfrac{8020}{12} \right\rfloor = 668

of them. Thus, the correct answer is D.

More practice

Concepts: least common multiple · multiple · counting integers in a range

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.