Okay, I have been at this for 4 hours and I have made no progress so any input on how to go about this would be appreciated.
I am given a random integer that is supposed to represent a number of basketballs.
If the integer is even, I can take away half of the balls.
If the number is divisible by 5, I can take away 42 balls.
And if the number is divisble by 3 or 4, I can take the last two digits multiplied together and subtract that from the number of balls.
The goal is to end up with 42 balls.
How would I go about calculating all possible cases where I can reach that goal?

I'm assuming you mean that you can apply these rules over and over until you potentially reach 42.
This is a problem with recursion. Your terminating condition is when the number of balls is less than or equal to 42. Every operation you describe reduces the number of balls and approaches the terminating condition.

Well, I was close earlier. Also, I believe there would need to be a check to make sure the lastDigit or secondToLast don't equal 0.

You only need to check the second to last, since if the last is 0 it is divisible by 2 (and 5).

Wrong.

Case: 60
0 == 60 % 2: valid(60/2)? 30 < 42, no
0 == 60 % 5: valid(60-42)? 18 < 42, no
0 == 60 % 3: 0 != secondtolast(60): valid(60-6*0)? Iunno, try valid(60).

Oh, wait, infinite recursion!

(In short, yes, check both the last and second to last digits.)

Oops, good point. I had assumed (that terrible word) the mod tests covered all matching cases without reading the code properly. The inner conditions do indeed let them through.

Is this supposed to be recursive rules or apply once rules?