Euclidean Division

Question

Let $n\in N-\{0,1\}$: Determine the rest of the euclidean division of $2^n-1$ over $2^{n-1}$

Could anyone give me some hints in order to approach this problem? thank you in advance.

Answer 1

You have that

$$2^n -1 = 2 \times 2^{n-1} -1 = 1\times 2^{n-1} + (2^{n-1}-1)$$

And $0 \leq (2^{n-1}-1) < 2^{n-1}$, so it is the rest of the division of $2^n -1$ by $2^{n-1}$