The number 119 has the following properties:

(a) division by 2 leaves a remainder of 1;

(b) division by 3 leaves a remainder of 2;

(c) division by 4 leaves a remainder of 3;

(d) division by 5 leaves a remainder of 4;

(e) division by 6 leaves a remainder of 5.

How many positive integers less than 2007 satisfy these properties?
Given positive integers $d$ and $r$, we say that $N$ divided by $d$ leaves a remainder $r$ if there exists an integer $n$ such that $N = n d + r$. If $M$ divided by $d$ also leaves the same remainder $r$, then there exists an integer $m$ such that $M = m d + r$. In this case, if $M > N$, it follows that $m = n + p$ for some integer $p$, and therefore,

$$
M = m d + r = (n + p) d + r = n d + r + p d = N + p d
$$

so that $M - N = p d$ is a multiple of $d$. The same applies if $M < N$.

Thus, the only positive integers less than 2007 with the same division properties as 119 are

$$
59, 119, 179, \ldots, 1979 (= 119 + 60 \times 31)
$$

for a total of 33 numbers.