a) Suppose that n and k are positive integers with n congruent to k mod 40.
prove that 2^n is congruent to 2^k mod 55.
b) Prove that for any integer, n
n^11 is congruent to n mod 33.
If then for some .
This means, .
Since by Euler's theorem.
Consider the group .b) Prove that for any integer, n
n^11 is congruent to n mod 33.
This group is isomorphic to .
Therefore, has exponent i.e. it means .
Thus, if it would mean so .
Now if then it means .
If there is nothing to prove.
To complete the proof note that if this is true by mods.
Thus, if for example then with relatively prime to .
But then .
And with that this completes the proof.