Prove that $a^{20} = 1 \bmod 440$ if $\gcd(a,440)=1$.

Question

Prove that $a^{20} = 1 \bmod 440$ if $\gcd(a,440)=1$.

So far I have that $\varphi(440)=160,$ so $a^{160}=1 \bmod 440$, but I am not sure how to get to $a^{20}$.

Answer 1

Hint: If $x$ and $y$ are coprime,

$$a^m\equiv 1\bmod x,$$

and

$$a^n\equiv 1\bmod y,$$

then

$$a^{\mathrm{lcm}(m,n)}\equiv 1\bmod xy.$$

Can you use this, combined with your knowledge of the prime factorization of $440$, to get the desired result?

(Note: This is heavily related to the Carmichael lambda function, which can be considered a "stronger version" of Euler's $\varphi(n)$.)