112. Make sure that the last digits of the numbers in the Fibonacci sequence $^{\circ}$ repeat periodically. What is the length of the period?

Ensure that the last digits of the numbers in the Fibonacci sequence $^{\circ}$ repeat periodically. What is the length of the period?
112. The last digits of the Fibonacci sequence $\varnothing^{0}$ themselves form a Fibonacci sequence in the sense of 10-arithmetic (sequence $\Phi_{10}^{0}$).

Let's write out the terms of this sequence:

$0,1,1,2,3,5,8,3,1,4,5,9,4,3,7$, $0,7,7,4,1,5,6,1,7,8,5,3,8,1,9$, $0,9,9,8,7,5,2,7,9,6,5,1,6,7,3$, $0,3,3,6,9,5,4,9,3,2,5,7,2,9,1$, $0,1,1,2,3,5, \ldots$

We notice that $c_{60}=0, c_{61}=1$. Therefore, if we discard the first 60 terms $\left(c_{0}, c_{1}, c_{2}, \ldots, c_{60}\right)$ of the sequence, we are left with a Fibonacci sequence starting with the numbers 0,1, i.e., the sequence $\Phi_{10}^{0}$ again. This means that $c_{62}=c_{2}, c_{63}=c_{3}, \ldots, c_{60+k}=$ $=c_{k}, \ldots$, i.e., the sequence $\Phi_{10}^{0}$ is periodic with a period length of 60.