## Online Course Discussion Forum

### 2021 8 Week Prep Course for the AMC 10 Question 1

2021 8 Week Prep Course for the AMC 10 Question 1

What's a quick way to solve this problem and other problems like this?

How I solved it was I looked at each exponent of 23 and tried to find a pattern but I would assume that would be too time-consuming on the actual AMC.

Re: 2021 8 Week Prep Course for the AMC 10 Question 1

Try using Euler's Theorem (the extension of Fermat's Little Theorem for when the mod is not prime) that says $$a^{\phi(n)} \equiv 1 \pmod{n},$$ where $\phi(n)$ is Euler's totient function (number of positive integers smaller than $n$ that are relatively prime with $n$) and $\gcd(a,n) = 1$.

Re: 2021 8 Week Prep Course for the AMC 10 Question 1

Got it, thank you!
Re: 2021 8 Week Prep Course for the AMC 10 Question 1

I'm also struggling with this problem. I can't seem to find a pattern by looking at the powers of 23 mod 100, or figure out how to use Euler's Theorem. I've tried putting in 100 and 10 for n and 23 for a but I can't figure out how to get the 2323 into the exponent. I'm not really familiar with this theorem and how to use it, so do you have any hints?

Re: 2021 8 Week Prep Course for the AMC 10 Question 1

Oh wait never mind I was able to find a pattern with the powers of 23, but I still don't know how to use Euler's Theorem for this problem.