physics
posted by s .
Let α⟩=1M√∑M−1j=0αjj⟩ and let β⟩=1M√∑M−1j=0βjj⟩ be its QFTM. Suppose we shift the superposition α⟩ to produce α′⟩=1M√∑M−1j=0αjj+1(modM)⟩, and let β′⟩=1M√∑M−1j=0βj′j⟩ be the QFTM of α′⟩. Derive an expression for β′j as a function of βj. You can use e, j, pi, and M in your response.

Got this result for above question:
exp(2ðji/M)âj
However it is graded as wrong, can someone help? 
exp(2piji/M)beta j

In this problem, we will carry out some steps of the quantum factoring algorithm for N = 15
(a) What is the period k of the periodic superposition set up by the quantum factoring algorithm if it chooses x = 2 ?
(b) Assume that we found this k using period finding algorithm. Use k to find a nontrivial square root of 1(mod 15) . Write your answer as an integer between 0 and 15.
(c) Then, the algorithm proceeds by computing gcd (x,y) for some integers x and y . List these two numbers separated by a comma.
If there are more than one correct solution, provide any one of them. 
WHAT IS THE ANSWER FOR:
Let α⟩=1M√∑M−1j=0αjj⟩ and let β⟩=1M√∑M−1j=0βjj⟩ be its QFTM. Suppose we shift the superposition α⟩ to produce α′⟩=1M√∑M−1j=0αjj+1(modM)⟩, and let β′⟩=1M√∑M−1j=0βj′j⟩ be the QFTM of α′⟩. Derive an expression for β′j as a function of βj. You can use e, j, pi, and M in your response.