Relative primes What is the number of integers between 1 and 60 that are relatively prime to 60? I know that the answer is 16, but how do I go about finding the relative primes using a quick process?

You can just use Euler's totient function. Given a prime $p$, $\phi(p^\alpha) = (p - 1)p^{\alpha - 1}$. The function is multiplicative, so $\phi(p^\alpha q^\beta) = \phi(p^\alpha) \phi(q^\beta)$.
Since $60 = 2^2 \times 3 \times 5$, you calculate $$\phi(60) = \phi(4) \phi(3) \phi(5) = 2 \times 2 \times 4 = 16.$$
To verify this answer, let's count up the primes from 7 to 59, and throw in there the odd composite numbers not divisible by 3 or 5, to get: 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59; this is a list of 16 integers.
I suppose either way is a quick enough process for small numbers. But if the numbers get just a little bit bigger, Euler's totient function is much faster: try for example to count up the integers between 1 and 5168743489 that are relatively prime to 5168743489.
