Street

Cliff · residue

Four operations on the ring.

+1 is the cycle. ×47 is the unit. ×47⁻¹ is extended Euclid. CRT splits and rebuilds Z₄₀ only — it does not reconstruct Z₁₂₅. Euler: if gcd(a,125)=1 then a¹⁰⁰≡1.

Outer Z₁₂₅ · inner Z₄₀ · click to index

  • 47 · 47⁻¹1 (mod 125)

    Extended Euclid: 47⁻¹ ≡ 8 (mod 125). Product must be 1.

  • CRT round-trip40/40

    split 47 → (7 mod 8, 2 mod 5), combine → 7. Reconstructs Z_40, not Z_125.

  • Euler a^{φ(125)}1 (mod 125)

    gcd(47,125)=1 so 47^{100} ≡ 1. φ(125)=100. Same order we walked for 47.

  • +1 then −147

    Cyclic shift on the outer ring. Inverse of +1 is +124. p₃=47, p₄=233.