What theory could I look at to predict what the last digit will be when 3 is raised to the power of 200?

Also - my question reads "are there any whole numbers that give the same last digit when they are raised to any power?" Where should I start my investigation here? Is there a theory that simplifies this?

Would appreciate any feedback - thanks.

Hello there:

Notice the pattern of the last digits of powers of 3:

3^{1}=3\\3^{2}=9\\3^{3}=27\\3^{4}=81\\3^{5}=243\\3 ^{6}=729\\3^{7}=2187\\3^{8}=6561

See the last digit pattern? Now, where does 200 fit in?

Since 4 divides evenly into 200, what should be the digit?

Your second question should be easy enough to work out too.

There are a whole load of numbers that always end in 0 when multiplied by anything, let alone itself.