1. ## Modulo 20 exponents?

Im trying to find 2018^2018 mod 20. Are there rules to help with this? I know a^b = c^b mod 20 if a=c mod 20, so you can do 18^2018 mod 20, but I dont know where to go from there.

2. ## Re: Modulo 20 exponents?

Originally Posted by Ilikebugs
Im trying to find 2018^2018 mod 20. Are there rules to help with this? I know a^b = c^b mod 20 if a=c mod 20, so you can do 18^2018 mod 20, but I dont know where to go from there.
For quick answers I go here

3. ## Re: Modulo 20 exponents?

Is there a way to prove the answer without using a calculator or online resource using rules for modular arithmetic?

4. ## Re: Modulo 20 exponents?

$$20=2^2\cdot 5$$

$$\phi(20)=8$$

This means:

$$\forall x,r \in \mathbb{Z}, r>0 , x^{8k+r} \equiv x^r \pmod{20}$$

So you have:

$$2018 \equiv 18 \pmod{20}$$

$$2018 \equiv 2\pmod{8}$$

So, this gives:

$$2018^{2018} \equiv 18^2 \pmod{20}$$

5. ## Re: Modulo 20 exponents?

Originally Posted by SlipEternal

This means:

$$\forall x,r \in \mathbb{Z}, r>0 , x^{8k+r} \equiv x^r \pmod{20}$$
Not true

Take for example $$x=2,k=1,r=1$$

6. ## Re: Modulo 20 exponents?

Originally Posted by Idea
Not true

Take for example $$x=2,k=1,r=1$$
It was late and I had a typo. I meant $r>1$