Editing Openai/6933ffb6-9ed8-8010-99bb-a5bc968f2d9b (section)

=== Prompt (exact): ===

: 

Correct answer:
807

Verification rubric:
* Pass if the model returns 807 as the remainder (or “the remainder is 807”) and gives the supporting CRT/modular-phi step (mod 8 = 7, mod 125 =  ??? leading to 807).
* Fail if it gives any other number or an answer without consistent modular reasoning.

Concise solution / explanation:
* Use CRT with 1000 = 8 × 125.
* Mod 8: 7² ≡ 1 (mod 8) so powers of 7 cycle length 2. 2025 ≡ 1 (mod 2) → 7^{2025} ≡ 7 (mod 8).
* Mod 125: φ(125)=100 so 7^{2025} ≡ 7^{25} (mod 125). Compute 7^{25} (or do repeated squaring mod125) → you find it ≡ 57 (mod 125). (You can verify by direct modular exponentiation).
* Solve x ≡ 7 (mod 8), x ≡ 57 (mod 125) → unique solution mod 1000 is 807. (Direct computation also gives 7^2025 mod 1000 = 807.)