Given $0 <(x, y) < z < 2^{64}$, How can I compute $\lfloor \frac{xy}{z} \rfloor$ using only 64-bit arithmetic operations?
Use two integers to represent one integer. Assume you are using c/c++. You can use a struct that has two long long unsigned integers to represent $x$, $y$ and $z$. So you can express $x = 2^{32}x_1+x_2$ with $x_1$ being a nonnegative integer smaller than $2^{32}$ and $x_2$ being a nonnegative integer not greater than $2^{32}$. I would assume you will know the rest.

Or you can choose to use some libraries or built-in classes. For example, you can use BigInteger class in Java.

(It goes without saying that you will not have these problems if you were using Python, Ruby, etc.)