Finding valid keys in a cryptosystem (m^k + mod 41)

Question

I'm studying for an exam I have tomorrow and could not find the answer to the following question:

For which of the following values of k is $$ E_{k}(m) = m^{k} mod 41 $$ a cipher over $$ Z_{41} $$ ?

And the possible values of k is: 3, 5, 7

Answer 1

If by a cipher you mean a bijection, then both $k=3$ and $k=7$ work because they are prime with $40=\phi(41)$, but $k=5$ isn't.

Answer 2

5 doesn't work because it divides 40 (which is phi(41)); so 5 has no multiplicative inverse mod 40. 3 and 7 are relatively prime to 40 so in each of these cases there is a multiplicative inverse (and therefore a bijection).