Saturday, 7 February 2026

Exercise (5.1).9

What is the probability that a number $m \in \{1, 2, 3, \ldots , 164\}$ is relatively prime to 164?


There are $\phi(164)$ numbers from 1 to 164, inclusive, that are coprime to 164.

Let's calculate this first

$$ \phi(64) = \phi(2^2 \times 41) = \phi(2^2) \times \phi(41) = 2^1(2-1) \times (41-1) = 80 $$

And so the probability is

$$ \frac{80}{164} = \frac{20}{41} \approx 0.49$$