1. **State the problem:** Find the remainder when $7^{222}$ is divided by 100. 2. **Formula and theorem:** We use Euler's theorem which states that if $a$ and $n$ are coprime, then $$a^{\phi(n)} \equiv 1 \pmod{n}$$ where $\phi(n)$ is Euler's totient function. 3. **Calculate $\phi(100)$:** Since $100 = 2^2 \times 5^2$, $$\phi(100) = 100 \times \left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{5}\right) = 100 \times \frac{1}{2} \times \frac{4}{5} = 40$$ 4. **Apply Euler's theorem:** Since $7$ and $100$ are coprime, $$7^{40} \equiv 1 \pmod{100}$$ 5. **Reduce the exponent modulo 40:** $$222 \equiv \cancel{40 \times 5}^{200} + 22 \equiv 22 \pmod{40}$$ 6. **Calculate $7^{222} \equiv 7^{22} \pmod{100}$:** We compute powers of 7 modulo 100 stepwise: $$7^1 = 7$$ $$7^2 = 49$$ $$7^4 = (7^2)^2 = 49^2 = 2401 \equiv 1 \pmod{100}$$ 7. Since $7^4 \equiv 1 \pmod{100}$, we write $$7^{22} = 7^{4 \times 5 + 2} = (7^4)^5 \times 7^2 \equiv 1^5 \times 49 = 49 \pmod{100}$$ **Final answer:** $$\boxed{49}$$