elementary number theory - Exponent of Prime in a Factorial

I was just trying to work out the exponent for $7$ in the number $343!$. I think the right technique is $$\frac{343}{7}+\frac{343}{7^2}+\frac{343}{7^3}=57.$$ If this is right, can the technique be generalized to $p$ a prime number, $n$ any positive integer, then the exponent of $p$ in $n!$ will be $$\sum_{k=1}^\infty\left\lfloor\frac{n}{p^k}\right\rfloor\quad ?$$ Here, $\lfloor\cdot\rfloor$ denotes the integer less than or equal to $\cdot$ .

Obviously the sum is finite, but I didn't know if it was correct (since its veracity depends on my first solution anyway).