1. Euler's theorem states that if $n$ and $a$ are coprime (i.e., their greatest common divisor is 1), then: $$a^{\varphi(n)} \equiv 1 \pmod{n}$$ where $\varphi(n)$ is Euler's totient function, which counts the positive integers up to $n$ that are coprime to $n$. 2. Let's understand this with an example: Find $3^{\varphi(10)} \pmod{10}$. 3. First, calculate $\varphi(10)$. The number 10 has prime factors 2 and 5, so: $$\varphi(10) = 10 \times \left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{5}\right) = 10 \times \frac{1}{2} \times \frac{4}{5} = 4$$ 4. Since $3$ and $10$ are coprime, Euler's theorem applies. Compute: $$3^{4} = 81$$ 5. Now find $81 \pmod{10}$: $$81 \div 10 = 8 \text{ remainder } 1$$ So, $$3^{4} \equiv 1 \pmod{10}$$ 6. This confirms Euler's theorem: $3^{\varphi(10)} \equiv 1 \pmod{10}$. Final answer: $$3^{4} \equiv 1 \pmod{10}$$