# Thread: Some application of Euler's Theorem probably

1. ## Some application of Euler's Theorem probably

$\text{Find the smallest }n \in \mathbb{N^+} \ni \text{ the last two digits of }3^n \text{ written in base }143 \text{ are }\{0,~1\}$

I know you want to show that

$3^n \equiv 1 \pmod{143^2}$

and that

$3^{\varphi(143^2)} \equiv 1 \pmod{143^2}$

but $\varphi(143^2)=17160$ is not the smallest value that satisfies this, 195 is.

How to derive 195?

2. ## Re: Some application of Euler's Theorem probably

if we can show that $a=5$ is the smallest positive integer such that

$3^a\equiv 1 \pmod {11^2}$

and $b=39$ is the smallest positive integer such that

$3^b\equiv 1 \pmod {13^2}$

then it will follow that $n=5*39=195$ is the smallest positive integer such that

$3^n\equiv 1 \pmod {143^2}$