Is Z an integer?

Hi, plugin approach is the best way to solve this question, but let's just look at the algebraic approach as well.

st.1
z^3= I, here I is an integer and can take both positive as well as negative values.

z= (I)^1/3
now z can be integer depending upon the value of I. if I = -3, then z = (-3)^1/3, which is clearly not an integer. if I =8, then (8)^1/3 =2, which is an integer. hence st. 1 alone is not sufficient

st.2
3z = k, here k is an integer, which can take both positive as well as negative values.
z = k/3
now depending upon the value of k, z can either integral or non-integral values.
e.g. if k=5 then z is not an integer
if k=-3, then z is an integer.

st.1 + st.2

z= (I)^1/3
z= k/3
therefore we have (I)^1/3 = k/3
3(I)^1/3 = k
right hand side = k which is an integer. therefore left hand side must also be an integer. i.e.3(I)^1/3 must be an integer. which is possible only if (I)^1/3 is an integer.
if (I)^1/3 is an integer, then z is also an integer.

hence z is an integer.

therefore C.