Diggits

Can you find what the last two digits of the number $4^{1999}$ are?

Two Many

What is the least square number which commences with six two's?

Power Crazy

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

Like Powers

Stage: 3 Challenge Level:

Investigate

$$1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n$$

and

$$2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$$

for different values of $n$.