1. A 100-digit number has the form $a=1777 \ldots 76$ (with 98 sevens in the middle). The number $\frac{1}{a}$ is represented as an infinite periodic decimal. Find its period. Justify your answer.
# Solution:

Notice that $a=16 \cdot 111 \ldots 11$. The last number $b$ consists of 99 ones. According to the rules for converting a common fraction to a decimal, the number $\frac{1}{b}=0,(00 \ldots 09)$. Its period is 99. Then, when multiplying this fraction by the number $\frac{1}{16}=0.0625$, the period will not change.

Answer: 99.