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?

Lastly - Well

Age 11 to 14 Challenge Level:

How can you simplify this so that it does not contain a power
raised to a power?

Can you find a pattern in the last two digits? How about
starting with some simple cases?