Is $0$ divisible by $0$? I know that there's resources out there but my professor told me that nothing is divisible by $0$ because you can't divide by $0$ which makes it a NaN

The class is related to programming we are defining a "divisible" function that takes two arguments and I have to return true or false, so that's the context. I tried to explain to him what the "Let $a,b∈\mathbb{Z}$. Then $a$ is divisible by $b$ if and only if there exists an integer $k$ such that $a=kb$" as one of the answerers said but the professor said that just because $a$ implies $b$ doesn't mean $b$ implies $a$.

Also, there is a strong hint that we are supposed to use a modulus standard built in function in the programming language and this mod function returns undefined if you ask it to do $0 \mod 0$ so there's also that.
Yes, it is. Formal definition of divisibility is the following:


  Let $a,b \in \mathbb{Z}$. Then $a$ is divisible by $b$ if and only if there exists an integer $k$ such that $a = kb$.


Now, in your case, we have $a = b = 0$, that is, $0 = k\cdot0$ and there exists infinitely many such $k$ so yes, $0$ is divisible by $0$.
If you use algebra, the answer is zero. If A / B = C then C times B = A. For this example, A, B, and C are all zero.

To sum it up in a word problem: A birthday cake will be divided evenly among children at a party. The party got cancelled. There was no birthday cake and no children showed up. How big was the slice of cake each child got? 
Answer: Zero grams.

Another example: An electrical wire contains zero watts and zero volts. How many amps of electricity is in the wire? Amps = Watts / Volts. 
Answer: Zero amps.