Understanding Co-prime Integers

Two integers are co-prime if their greatest common divisor (GCD) is 1. We need to find the count of positive integers $k$ such that $1 \le k < 30$ and $\gcd(k, 30) = 1$.

This count is given by Euler's totient function, $\phi(n)$, which calculates the number of positive integers less than or equal to $n$ that are co-prime to $n$. Since $\gcd(30, 30) = 30 \ne 1$, the number of integers $k$ such that $1 \le k < 30$ and $\gcd(k, 30) = 1$ is the same as $\phi(30)$.

Calculating Co-prime Count using Euler's Totient Function

First, find the prime factorization of 30.

$ 30 = 2 \times 3 \times 5 $

The formula for Euler's totient function $\phi(n)$ is:

$ \phi(n) = n \prod_{p|n} \left(1 - \frac{1}{p}\right) $

where the product is taken over the distinct prime factors $p$ of $n$.

For $n=30$, the distinct prime factors are $p_1 = 2$, $p_2 = 3$, and $p_3 = 5$.

Calculate $\phi(30)$:

$ \phi(30) = 30 \times \left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right) \times \left(1 - \frac{1}{5}\right) $

$ \phi(30) = 30 \times \left(\frac{1}{2}\right) \times \left(\frac{2}{3}\right) \times \left(\frac{4}{5}\right) $

Simplify the expression:

$ \phi(30) = \frac{30 \times 1 \times 2 \times 4}{2 \times 3 \times 5} = \frac{240}{30} = 8 $

Thus, there are 8 positive integers less than 30 that are co-prime to 30.