# Prove that if $\gcd(a,480)=1$, then $a^{16} ≡ 1 \pmod{480}$. [duplicate]

I would like to prove it using Euler's Theorem, which states that if $$\gcd(a,n)=1$$,then $$a^{\phi(n)}=1 \pmod{n}$$. Then, $$480$$ should be factored, but I am unsure how to proceed with the proof. Thanks in advance.

• Did you at least calculate $\phi(480)$? Nov 10 '20 at 1:57
• Yes! $\phi(480)=128$, but I do not understand how that helps. Nov 10 '20 at 1:59
• @qt_314, s use mathworld.wolfram.com/CarmichaelFunction.html Nov 10 '20 at 2:08
• Same proofs as in the dupe work here. Nov 10 '20 at 2:12
• @BillDubuque Thank you so much! I apologize for posting a duplicate question. Nov 10 '20 at 2:14

Hint: $$480 = 2^5 \cdot 3 \cdot 5$$. By the Chinese remainder theorem, we see that $$a^k \equiv 1 \pmod {480}$$ if and only if we have $$\begin{cases} a^k \equiv 1 \pmod{2^5},\\ a^k \equiv 1 \pmod 3,\\ a^k \equiv 1 \pmod 5. \end{cases}$$