Hi didieravoaka,

This prompt is an example of a 'Symbolism' question - the prompt 'makes up' a math symbol, tells you what it means and asks you to perform a calculation using it. Based on the information in the prompt, we're told that....

32~ is the remainder when (32-1)! is divided by 32.

Now, there's no way that the GMAT would expect you to calculate the value of 31!, so we have to think in terms of what 31! actually is.

Here's an example that's a bit easier to deal with:

4! = (4)(3)(2)(1) = 24

What numbers divide EVENLY into 24?

1, 2, 3, 4, 6, 8, 12 and 24

You can clearly see why 1, 2, 3 and 4 divide in - they're in the 'chain' of numbers that are multiplied together.
6 divides in because (2)(3) = 6 - and you can see the (2) and the (3) in the 'chain'
Similarly, 8, 12 and 24 are also 'combinations' of the numbers in the 'chain', so they divide evenly in too.

31! has LOTS of numbers in it, so it's evenly divisibly by LOTS of different integers. If you were to write out 31!, you would see a (2) and a (16). This means that (2)(16) = 32 divides evenly into 31!, so there will be a remainder of 0.

Final Answer: A

GMAT assassins aren't born, they're made,
Rich