1. The problem asks to find which number divides the expression $$1^n + 2^n + 3^n + 4^n$$ given that $$n$$ is not divisible by 4. 2. We analyze the behavior of $$1^n + 2^n + 3^n + 4^n$$ under modulo arithmetic for the options: 3, 5, 7, and 9. 3. Check modulo 3: - $$1^n \equiv 1 \pmod{3}$$ always. - $$2^n$$ modulo 3 cycles as $$2, 1, 2, 1, \dots$$ based on whether $$n$$ is odd or even. - $$3^n \equiv 0 \pmod{3}$$. - $$4^n = (1^n \cdot 4^n) \equiv 1^n \equiv 1 \pmod{3}$$ because 4 ≡ 1 mod 3. Summing: $$1 + 2^n + 0 + 1 \pmod{3}$$ depends on $$2^n\pmod{3}$$, which is not always zero, so not divisible by 3. 4. Check modulo 5: - $$1^n \equiv 1 \pmod{5}$$. - Using Euler's theorem for 2 and 3 modulo 5: For $$2^n$$, $$2^4 \equiv 1 \pmod{5}$$, pattern repeats every 4. Also $$3^n$$ cycles mod 5 with period 4. - Since $$n$$ is NOT divisible by 4, the sums of $$2^n + 3^n + 4^n$$ modulo 5 cycle. - Testing values confirms $$1^n + 2^n + 3^n + 4^n \equiv 0 \pmod{5}$$ for $$n$$ not divisible by 4. 5. Check modulo 7 and 9 shows no consistent divisibility. **Conclusion:** The sum $$1^n + 2^n + 3^n + 4^n$$ is divisible by 5 when $$n$$ is not divisible by 4. **Final answer: 5**