Code to Zero

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

N000ughty Thoughts

How many noughts are at the end of these giant numbers?

Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Obviously?

Stage: 4 and 5 Challenge Level:

Consider separately when $N = 1^n + 8^n - 3^n - 6^n$ is divisible by $2$ and when it is divisible by $3$.