I'm trying to prove that$F_{5n+3}\text{mod}10 = L_{n}\text{mod}10$.
I rearranged it into a more solvable form of $F_{5n+3}-L_n = 10k$ (because if two numbers end in the same digit, their difference must be 10 and vice versa).
The problem is when I try to convert the Lucas numbers to Fibonacci numbers using almost any method, I get $F_{5n+3} - F_{n+1} - F_{n-1} = 10k$ or something similar. I need to find some way to get the coefficients to be equal so I can recombine them into simpler terms.
I'd appreciate any advice.
Hint: Both Lucas numbers and Fibonacci numbers have periodic residues modulo any number. Now just find the periods and residues for both...