19th Eötvös 1912

Prove that 5ⁿ + 2 3^n-1 + 1 = 0 (mod 8).

Solution

If n is even, 5ⁿ = 1 mod 8, 3^n-1 = 3 mod 8, so expression = 1 + 6 + 1 = 0 mod 8. If n is odd, then 5ⁿ = 5 mod 8, 3^n-1 = 1 mod 8, so expression = 5 + 2 + 1 = 0 mod 8.

19th Eötvös 1912

© John Scholes
jscholes@kalva.demon.co.uk
29 Oct 2003
Last corrected/updated 29 Oct 03