Calculate 2^100 in ZZ11?

Question

Calculate 2^100 in ZZ₁₁. 

 

This is a Linear Algebra problem.

 

Here's my work:

2^5=32=-1 mod 11

2^10=1 mod 11

2^100=1 mod 11

So I think the answer is 1 since that's the remainder. What's the answer? 

Answer 1

I did it with patterns and came to the same conclusion,

 

2^1 = 2 mod 11

2^2 = 4 mod 11

2^3 = 8 mod 11

2^4 = 5 mod 11

2^5 = 10 mod 11

2^6 = 9 mod 11

2^7 = 7 mod 11

2^8 = 3 mod 11

2^9 = 6 mod 11

2^10 = 1 mod 11

2^11 = 2 mod 11

2^12 = 4 mod 11

2^13 = 8 mod 11

 

So you can see that the pattern repeats after every 10...so 2^20, 2^30, 2^40.....2^100 are all 1 mod 11.

 

Not elegant but it does confirm your answer.