Hi guys! I need help understanding what the remainder would be for problems such as the one linked below. I understand how to find the remainder of 7^1, 7^2, etc. And I know why we are finding those remainders (to find a pattern) but I am unsure of how to find the remainder for 7^31. How would I apply the pattern to the question? If someone can break it down for me, I would appreciate it.

The idea here is pattern recognition. That is, the remainders of

7^1, 7^5, 7^9...

are all the same.

Try that to see if you can find out what the remainder of 7^{31} is.