A friend of mine recently told me that it is not possible to perfectly divide a cake in three pieces because 1/3 is an repeating decimal. Now, this is clearly a silly statement as 0.33333... is an repeating decimal but it is a real number nonetheless. My friend is right in the sense that is not possible to divide a cake in any "perfect" way, e.g. in two halves; more precisely, it is not possible to measure anything "perfectly" so that you will never know if you division was actually perfectly accurate.
I wanted to prove my friend wrong using the following argument: let's imagine you are right, you cannot divide a cake in three pieces but you can divide it in two pieces. Then you can divide those two pieces in two pieces and so on. At one point, you are going to have x pieces of cakes where x is divisible by three. Thus, your original statement must be false. While I was thinking this, however, I noticed that there is no number $2^n$ that can be divided by three without a remainder.
Has this been proved (I guess so)? Is there any intuitive explanation?