Question

What is the remainder when 2 to the power of 2009 and 2009 squared are added together and divided by 13?

Answer 1

The answer is 3. Do you need a mathematical explanation?

Answer 2

I think I need an explanation.

Answer 3

hang on a sec

Answer 4

2009^2 will have the last 2 digits 81.
2^n will have last digit:

2 if n mod 4 = 1
4 if n mod 4 = 2
6 if n mod 4 = 3
8 if n mod 4 = 0

2009 mod 4 = 1 so the last digit will be 2

(......2 + .....1) = ...... 3
(.....3) mod 13 could possibly be 3

This is not definitely true though, if (the number was ....13) for example, then the answer would be 0.

You would have to look into the patterns of the last 2 digits of 2^n. There is a pattern I can tell you that.

From the pattern you could tell if it would be 3 or 0.

However, I just put it into wolframalpha.com to get the result of 3

Answer 5

Actually I am wrong. It depends on much more than the first two numbers, because you are dividing by 13. So I don't have an explanation on this one. Sorry