Question

1. Let S= 1^1 + 2^2+3^3+.....+2010^2010. . What is the remainder when S is divided by 2?
2. Find the number of odd coefficients in expansion of (x + y)^2010.
3. Find all prime numbers p and integers a and b (not necessarily positive) such that p^a + p^b
is the square of a rational number.