Let *N* be a positive power of 2, *N*=2^{n}.
Then given any number *a* between 0 and *N*-1, it can be written
as

with the coefficients a_{k} equals to either 0 or 1
(they are the bits of the binary representation of *a*).
Consider the product

where the numbers A_{k} are less than one, and

ie their binary representation is 0.a_{k}...a_{0}.

Prove that the above product is equal to *N* if *a*=0,
and is zero otherwise.

**Hint**

Consider the sum Sum_{b=0..N-1} exp( 2 pi i a b / N ).
This result is used in the quantum Fourier transform.

Marco Corvi - 2004