1. Determine the number of pairs of positive integers $(a, b)$, with $a \leq b$, for which $\operatorname{lcm}(a, b)=2004$.

$\operatorname{lcm}(a, b)$ denotes the least common multiple of $a$ and $b$.

Example: $\operatorname{lcm}(18,24)=72$.
1. Factorize 2004: $2004=2 \times 2 \times 3 \times 167$. Since the number of pairs does not depend on the value of the prime numbers, but only on the number of prime numbers and their exponents, we will write $2004 p^{2} q r$ instead of $2004 p^{2} q r$. The pairs are ordered as follows: the number with the most prime factors as the first number, and in case of an equal number, alphabetically. Given values of $p, q$ and $r$, the found pairs can then be directly written as $(a, b)$ with $a \leq b$.

To start, we have all pairs consisting of $p^{2} q r$ and one of its divisors.

There are 12 of these, [ $\left.p^{2} q r, p^{2} q, p^{2} r, p q r, p^{2}, p q, p r, q r, p, q, r, 1\right]$.

Then there are pairs with the first number being $p^{2} q$ and the second number thus

a multiple of $r$: $\left(p^{2} q, r\right),\left(p^{2} q, r p\right),\left(p^{2} q, r p^{2}\right),\left(p^{2} q, r q\right)$.

Next, pairs with the first number being $p^{2} r: \left(p^{2} r, q\right),\left(p^{2} r, q p\right),\left(p^{2} r, q r\right)$.

Then pairs with the first number being $p q r$: (pqr, $\left.p^{2}\right),\left(p q r, p^{2} q\right),\left(p q r, p^{2} r\right)$.

And finally a few pairs with the first number being $p^{2}:\left(p^{2}, q r\right)$

So there are a total of $12+4+3+3+1=23$ pairs.