Title: Openai/6966800a-dc98-800e-af77-c82407982594 - freem

URL Source: https://freemwiki.com/wiki/Openai/6966800a-dc98-800e-af77-c82407982594

Markdown Content:
_This conversation was exported from ChatGPT_[[1]](https://freemwiki.com/wiki/Openai/6966800a-dc98-800e-af77-c82407982594#cite_note-1)

## gpt-5-2-instant[[edit](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&veaction=edit&section=1 "Edit section: gpt-5-2-instant") | [edit source](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&action=edit&section=1 "Edit section's source code: gpt-5-2-instant")]

### User: GPT, 이 문제를 풀어줄래?: $$Fx x^{2029} - 3x^{2028} + 4x^{2027} - 5x^{2026} + 6x^{3} - 7x + 8$$을 $x^4 - 1$로 나눈 나머지 $Rx$를 구해줘.[[edit](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&veaction=edit&section=2 "Edit section: User: GPT, 이 문제를 풀어줄래?: $$Fx  x^{2029} - 3x^{2028} + 4x^{2027} - 5x^{2026} + 6x^{3} - 7x + 8$$을 $x^4 - 1$로 나눈 나머지 $Rx$를 구해줘.") | [edit source](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&action=edit&section=2 "Edit section's source code: User: GPT, 이 문제를 풀어줄래?: $$Fx  x^{2029} - 3x^{2028} + 4x^{2027} - 5x^{2026} + 6x^{3} - 7x + 8$$을 $x^4 - 1$로 나눈 나머지 $Rx$를 구해줘.")]

GPT, 이 문제를 풀어줄래?: $$F(x) = x^{2029} - 3x^{2028} + 4x^{2027} - 5x^{2026} + 6x^{3} - 7x + 8$$을 $x^4 - 1$로 나눈 나머지 $R(x)$를 구해줘. ($\deg R \le 3$)

### Assistant: 핵심 아이디어는[[edit](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&veaction=edit&section=3 "Edit section: Assistant: 핵심 아이디어는") | [edit source](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&action=edit&section=3 "Edit section's source code: Assistant: 핵심 아이디어는")]

핵심 아이디어는

x4≡1(modx4−1)x^4 \equiv 1 \pmod{x^4-1}x4≡1(modx4−1) 을 이용해 지수들을 4로 나눈 나머지만 남기는 것입니다. 나머지 R(x)R(x)R(x)는 차수가 3 이하인 다항식입니다.

#### 1️⃣ 각 항을[[edit](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&veaction=edit&section=4 "Edit section: 1️⃣ 각 항을") | [edit source](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&action=edit&section=4 "Edit section's source code: 1️⃣ 각 항을")]

𝑥 4 − 1 x 4 −1로 정리

지수들을 4로 나눈 나머지를 계산하면

*   2029≡1(mod4)2029 \equiv 1 \pmod{4}2029≡1(mod4) → x2029≡xx^{2029} \equiv xx2029≡x
*   2028≡0(mod4)2028 \equiv 0 \pmod{4}2028≡0(mod4) → x2028≡1x^{2028} \equiv 1x2028≡1
*   2027≡3(mod4)2027 \equiv 3 \pmod{4}2027≡3(mod4) → x2027≡x3x^{2027} \equiv x^3x2027≡x3
*   2026≡2(mod4)2026 \equiv 2 \pmod{4}2026≡2(mod4) → x2026≡x2x^{2026} \equiv x^2x2026≡x2
*   x3x^3x3 그대로
*   xxx 그대로
*   상수항 그대로

따라서

F(x)≡x−3⋅1+4x3−5x2+6x3−7x+8\begin{aligned} F(x) &\equiv x - 3\cdot 1 + 4x^3 - 5x^2 + 6x^3 - 7x + 8 \\ \end{aligned}F(x)​≡x−3⋅1+4x3−5x2+6x3−7x+8​

#### 2️⃣ 같은 차수끼리 정리[[edit](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&veaction=edit&section=5 "Edit section: 2️⃣ 같은 차수끼리 정리") | [edit source](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&action=edit&section=5 "Edit section's source code: 2️⃣ 같은 차수끼리 정리")]

*   x3x^3x3: 4x3+6x3=10x34x^3 + 6x^3 = 10x^34x3+6x3=10x3
*   x2x^2x2: −5x2-5x^2−5x2
*   xxx: x−7x=−6xx - 7x = -6xx−7x=−6x
*   상수: −3+8=5-3 + 8 = 5−3+8=5

#### 3️⃣ 최종 나머지[[edit](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&veaction=edit&section=6 "Edit section: 3️⃣ 최종 나머지") | [edit source](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&action=edit&section=6 "Edit section's source code: 3️⃣ 최종 나머지")]

R(x)=10x3−5x2−6x+5\boxed{R(x) = 10x^3 - 5x^2 - 6x + 5}R(x)=10x3−5x2−6x+5​ 차수도 333 이하로 조건을 정확히 만족합니다.

## References[[edit](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&veaction=edit&section=7 "Edit section: References") | [edit source](https://freemwiki.com/index.php?title=Openai/6966800a-dc98-800e-af77-c82407982594&action=edit&section=7 "Edit section's source code: References")]

1.   [↑](https://freemwiki.com/wiki/Openai/6966800a-dc98-800e-af77-c82407982594#cite_ref-1 "Jump up")["다항식 나머지 구하기"](https://chatgpt.com/share/6966800a-dc98-800e-af77-c82407982594). ChatGPT. Retrieved 2026-01-14.