1. Problem: Find the remainder when $7^{5284}$ is divided by 5. 2. Formula: Use modular arithmetic and Euler's theorem or Fermat's little theorem. 3. Since 5 is prime, Euler's totient function $\\phi(5) = 4$. 4. By Fermat's little theorem, $7^4 \equiv 1 \pmod{5}$. 5. Find $5284 \mod 4$: $$5284 \div 4 = 1321 \text{ remainder } 0$$ 6. So, $$7^{5284} \equiv (7^4)^{1321} \equiv 1^{1321} \equiv 1 \pmod{5}$$ 7. Final answer: The remainder is 1.