Day of the Week Puzzle: Solving with Modular (Clock) Arithmetic

[Maw615]

I have this question and you need to use Modular (Clock) Arithmetic to solve it. If there is any work I need to know that. If you don't know the answer, is there anyway you can just tell me the process of how to do it? Thank you very, very much! Here is the problem:

What day of the week will 4^1000 days from a Sunday?

 

[Andre]

Have you considered that the week always repeats itself after seven days. Seven days after Sunday it's Sunday again, also after 7000000000 days. Consequently 700000000000000002 days from a Sunday it will be Tuesday. Would that help?

 

[Cantari]

Well after looking at this, I noticed a pattern. When you square 4 the first time, the remainder of that number divided by 7 is 2. when you cube 4 the remainder is 1. When you put it to the 4th the remainder is 4, then this pattern repeats. So then since it is to the 1000, it would be 4 days after sunday.

 

[Maw615]

Thank you very, very much for your help!