How many numbers from 1 to 30 are coprime with 30?
8
The count of numbers from 1 to n that are coprime with n is Euler's totient \(\phi(n) = n\left(1 - \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)\cdots\) over the distinct prime factors of n.
30 = 2 × 3 × 5, so \(\phi(30) = 30 \times \frac{1}{2} \times \frac{2}{3} \times \frac{4}{5} = 8\).
Check: listing them, the numbers from 1 to 30 with no factor 2, 3 or 5 are 1, 7, 11, 13, 17, 19, 23, 29, which is exactly 8 numbers.
Hence, 8.
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