Show that $2^{84} \pmod{377} \equiv 1 $

Question

$\varphi(377)=336$ and $gcd(377,2)=1$ so $2^{336} \equiv1\pmod{334}$

$84\cdot4=336$

I know that:\begin{cases} & 2^{84}\pmod{13}\Rightarrow2^{12}\equiv 1\pmod{13}\Rightarrow2^{84}\equiv 1\pmod{13}\\ &2^{84}\pmod{29}\Rightarrow2^{28}\equiv 1\pmod{13}\Rightarrow2^{84}\equiv 1\pmod{29}\end{cases}

How do I make the link to: $2^{84} \pmod{377}$ ?

Thanks.

Answer 1

The Chinese Remainder Theorem guarantees that there is exactly one $x$ satisfying $x\equiv 1 \mod 29$ and $x\equiv 1 \mod 13$ so since $1$ satisfies, you are done.

Answer 2

The Chinese remainder theorem says that $2^{84}\equiv 1\pmod{13}$ and $2^{84}\equiv 1\pmod{29}$ together implies $2^{84}\equiv 1\pmod{337}$.