Cannot find length of repeating block in decimal expansion for $\frac{17}{78}$ I am trying to find the length of of the repeating block of digits in the decimal expansion of $\frac{17}{78}$.
On similar problems, that has not been an issue. Take for instance $\frac{17}{380}$. My usual approach would be to calculate $\Phi (380) = \Phi(4)*\Phi(5)*\Phi(19)=2*4*18=144$, then test $10^{\Phi(each factor)} \equiv 1 \pmod{380}$. No $\Phi(factor)$ passes, so the highest, $\Phi(19) = 18$, is the length of the repeating block of digits.
But that does not work for $\frac{17}{78}$. I know from checking on my calculator that the length is 6, but there is no factor of 78 such that $\Phi(factor)=6$.
What makes this problem different and what method do I use to find the length of its repeating block?

$$17\equiv-1\pmod3\implies17^2\equiv1\pmod3\implies\text{ord}_3{17}=2$$
$$17\equiv4\pmod{13}\implies17^2\equiv16\equiv3,17^3\equiv51\equiv-1\implies\text{ord}_{13}{17}=6$$
So, we can derive ord$_{39}17=6$
We can prove


*

*$\lambda(n)$ always divides $\phi(n)$

*ord$_na$  always divides $\lambda(n)$ for $(a,n)=1$

