z^n = 1.

from 1. n is not zero.
from 2. z>0; z=1,2,3,4,5......

if, z = 2,3,4,5.. in this case z^n can't be 1. as the value comes up more than or less than 1.

As it is mentioned in the question Z^n = 1. in that case z =1. Hence, by using both the equations we can find out the value of z.

Z^n = 1

Before we get to the statements. Understand that Z^n = 1 means z has a magnitude of 1 but we don't know its sign. Could be positive or negative.

example (-1)^2=1 or (1)^2=1 or (-1)^10=1 etc. So we basically have different possible values of 1. If n=0 then whatever the value of z, z^n=1

(1)If n is non zero, this doesn't really help us. It only repeats a part of what I've mentioned above. z could be +1 or -1. INSUFFICIENT.

(2)About this statement. I'm sure u remember one important funda, Any number raised to zero equals 1

z could be 1,9,42,10^7 and if n=0 then z^n=1
Also, it could also be that 1^1=1 In which case z=1. Since we get conflicting answers this statement is INSUFFICIENT.

Combining both we get n is non zero and z is +ve . The only possible value for z in this case is 1. Coz 1 raised to the power of a +ve or -ve number is going to equal 1. Hence C

thanks Knight247 !!
I missed the part where z cud be (-1) and n = even and still z^n = 1