RSA: Given two very large prime $p,q$ and $n=p\;q$, how to quickly find $e$ such that $\gcd(\varphi(n),e)=1$? I can imagine that finding $p,q,n$ are easy, but my text book also says the $e$ can be found rapidly, so how does modern RSA find this $e$?
Normall practice is to pick public exponent as a small prime(preferably with few set bits in binary representstion). If the public exponent is prime we only need to verify it doesn't devide $\phi(n)$.