Solution:
First, find $N = \operatorname{lcm}(20, 18, 2018)$.

Prime factorizations:
$20 = 2^2 \times 5$
$18 = 2 \times 3^2$
$2018 = 2 \times 1009$

So,
$N = \operatorname{lcm}(2^2 \times 5, 2 \times 3^2, 2 \times 1009)$

Take the highest power of each prime:
- $2^2$ (from $20$)
- $3^2$ (from $18$)
- $5$ (from $20$)
- $1009$ (from $2018$)

Thus,
$N = 2^2 \times 3^2 \times 5 \times 1009$

Now, the number of positive integers less than $N$ and relatively prime to $N$ is $\varphi(N)$, where $\varphi$ is Euler's totient function.

$N = 2^2 \times 3^2 \times 5 \times 1009$

So,
$\varphi(N) = N \left(1 - \frac{1}{2}\right) \left(1 - \frac{1}{3}\right) \left(1 - \frac{1}{5}\right) \left(1 - \frac{1}{1009}\right)$

Calculate step by step:

$N = 4 \times 9 \times 5 \times 1009 = 36 \times 5 \times 1009 = 180 \times 1009 = 181620$

Now,
$\varphi(N) = 181620 \times \frac{1}{2} \times \frac{2}{3} \times \frac{4}{5} \times \frac{1008}{1009}$

Calculate each step:

$181620 \times \frac{1}{2} = 90810$
$90810 \times \frac{2}{3} = 60540$
$60540 \times \frac{4}{5} = 48432$
$48432 \times \frac{1008}{1009}$

Now, $48432 \times 1008 = 48831056$

Divide by $1009$:
$\frac{48831056}{1009} = 48408$

So, the answer is $48408$.

Final Answer:

$\boxed{48408}$