1. Let's state the problem: We want to prove or use Euler's theorem, which states that for any integer $a$ and $n$ that are coprime (\gcd(a,n)=1), $$a^{\phi(n)} \equiv 1 \pmod{n}$$ where $\phi(n)$ is Euler's totient function, representing the number of integers up to $n$ that are coprime with $n$. 2. First, verify that $a$ and $n$ are coprime (i.e., their greatest common divisor is 1). 3. Calculate $\phi(n)$, the Euler totient function value for $n$. 4. According to Euler's theorem, raise $a$ to the power of $\phi(n)$. 5. By modular arithmetic, show that: $$a^{\phi(n)} \equiv 1 \pmod{n}$$ 6. This means when $a^{\phi(n)}$ is divided by $n$, the remainder is 1. Therefore, Euler's theorem holds for the given integers $a$ and $n$.