Let a_n = (10^n-1)/9. 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?

Question

thomas5267 · Answer

That looks very familiar.  I think I saw this on this website today.

anonymous · Answer

gr8, can you please copy and paste that will hel

thomas5267 · Answer

I have no idea where the thread went.

thomas5267 · Answer

$$
a_1=1\
a_2=11\
a_3=111
$$

thomas5267 · Answer

It seems that the $d_n=1$ but I can't prove it.

anonymous · Answer

if that's the right ans

thomas5267 · Answer

@ganeshie8

anonymous · Answer

ok.. got it it right thanks

thomas5267 · Answer

AH!
$$
10^{n+1}-1-10(10^n-1)=r\
10^{n+1}-1-10^{n+1}+10=r\
r=9
$$
The greatest common divisor of $10^{n+1}-1$ and $10^n-1$ is 9.  Hence, the greatest common divisor of $\dfrac{1}{9}\left(10^{n+1}-1\right)$ and $\dfrac{1}{9}\left(10^n-1\right)$ is 1.

ganeshie8 · Answer

Nice :)