jmc

We are interested in how many numbers among $7,8,9,\dots ,59$ are relatively prime to 15.

First, we count how many numbers among $1,2,3,\dots ,60$ are relatively prime to 15. Note that $15=3\cdot 5$. Among these 60 numbers, $60/3=20$ are multiples of 3, $60/5=12$ are multiples of 5, and $60/15=4$ are multiples of 15. We can take 60, and subtract 20 and 12, but we have subtracted the multiples of 15 twice. Therefore, among the 60 numbers, there are $60-20-12+4=32$ numbers that are relatively prime to 15.

Going back to the set $7,8,9,\dots ,59$, we must account for the numbers 1, 2, and 4 that are relatively prime to 15. Thus, the answer is $32-3=\boxed{29}$.