Question: prove n^2+1 is not divisible by 4?
Answer: Simply: If n is odd, n^2 is odd so n^2 + 2 must be odd. If n is even, n^2 is a multiple of 4 so n^2 + 2 cannot be a multiple of 4 (will have remainder 2). If n is pair ( = 2m ) then n^2+2 = 4m^2 + 2; thus not divisible by 4.