jmc

Question

Let a_n = 10^n-19. Define d_n to be the greatest common divisor of a_n and a_n+1. What is the maximum possible value that d_n can take on?

Accepted Answer

From the Euclidean Algorithm, (a_n, a_n+1) = (a_n, a_n+1 - 10a_n). We compute a_n+1 - 10a_n = 10^n+1-19 - 10^n+1-109 = 1. Therefore, a_n+1 and a_n share no common factors and d_n is always 1.