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69ⁿ mod 420
69⁶⁹ ≡ 69
69³ ≡ 69 (mod 420). Every odd positive power of 69 is 69 modulo 420. Every even positive power is 141. Including 69⁶⁹ ≡ 69 (mod 420).
Raise 69 to an odd power. Divide by 420. The remainder is 69. Square first and it is 141. The cycle locks. So 69 to the 69th is 69 again, modulo 420. Exact. Not rare: 81 residues out of 420 satisfy x³ ≡ x (mod 420). 69 is one of them. The pretty reason is the consecutive triple 68 × 69 × 70, which already holds every prime-power in 420.
What it supports
Exact integer arithmetic. 69³ − 69 = 328440 = 782 × 420. 81 of 420 residues satisfy the cubic. Nearby 419 and 421 do not give this for 69.
What it does not
Not the Moon ratio. Not a% of b. Not 69 days after 4/20.
Calculation
69² = 4761 ≡ 141 (mod 420). 69³ ≡ 69×141 = 9729 ≡ 69 (mod 420). 69³ − 69 = 328440 = 782×420. 69^{2k+1} ≡ 69; 69^{2k} ≡ 141.
Integer congruence. 420 = 2²×3×5×7. 69=3×23. 69(69−1)(69+1)=69×68×70 supplies 2,3,5,7.
ExactnessExact
Form420
Unitmod
Strengthreproducible
Hung9 September 2026