4) How many multiples of 3 are there that are greater than or equal to 2000 and less than or equal to 4000?
(A) 666,
(B) 667,
(C) 668,
(D) 669,
(E) 670.
## 4. The answer is $(\mathbf{B})$.

The largest multiple of three less than 2000 is 1998. The subsequent multiples of three can be written as $a_{k}=1998+3 k$ with $k$ a positive integer. For example, for $k=1$ we get $a_{1}=2001$, the smallest of the numbers sought. It is necessary to determine the largest value of $k$ for which $a_{k}<4000$, that is, for which $1998+3 k<4000$.

The inequality is satisfied for $k<\frac{4000-1998}{3}$, that is, for $k<\frac{2002}{3}=667+\frac{1}{3}$ but, since $k$ is a positive integer, this is equivalent to saying $k \leq 667$. The number of multiples of 3 between 2000 and 4000 is therefore 667.