Problem:
Given that i2=−1, for how many integers n is (n+i)4 an integer?
Answer Choices:
A. none
B. 1
C. 2
D. 3
E. 4
Solution:
Since i2=−1,
(n+i)4=n4−6n2+1+i(4n3−4n).
This is real if and only if 4n3−4n=0. Since 4n(n2−1)=0 if and only if n=0,1,−1, there are only three values of n for which (n+i)4 is real; (n+i)4 is an integer in all three cases.