In Z5 compute (a+b)^5?

Answer 1

(a+b)^5 =
a^5 + 5a^4b + 10 a^3b^2 + 10 a^2b^3 + 5ab^4 + b^5 =
a^5 + b^5 in Z5

Answer 2

In Z_5 all terms in the expansion of (a+b)^5
except the first and last are multiples of 5.
So the answer is a^5 + b^5.
More generally, this holds for any prime p.
In Z_p all terms of the expansion of (a+b)^p
except the first and last are multiples of p,
so (a+b)^p = a^p + b^p in Z_p