1. Notice that 10 = 2*5 which implies that 10^n = 2^n * 5^n. Therefore 2^n can never equal to 10^n for any value of n.
2. Let a and b bet real numbers.
Notice that when a is divided by (a+b), the remainder is –b. This implies that when a^n is divided by (a+b), the remainder is (-b)^n. If n is odd, the preceeding statement is equivalent to saying that a^n +b^n is divisible by (a+b) for odd n.
It follows that 2^n + 3^n is divisible by 5 for odd n.
3. Notice that, an odd integer raised to any power is odd while an even number raised to any power is even and that if two numbers have the same parity then the sum is even otherwise it is odd.
This leads us to conclude that 1^n + 2^n + 3^n + 4^n is even because 1^n + 3^n is even and so is 2^n + 4^n . the other cases follow from similar reasoning