Question 134
Determine the prime numbers $p$ such that the decimal representation of the fraction $\frac{1}{p}$ has period of length 5.
Remark: If the decimal representation of a number has a sequence of digits that repeats periodically, the length of the shortest repeating sequence of digits is the length of the period of the decimal representation. For example, $61 / 495=0.1232323 \ldots$; although 23 and 2323 are sequences of digits that repeat in the decimal representation, the length of the shortest one is 2 and this is the length of the period.
Remark: If the decimal representation of a number has a sequence of digits that repeats periodically, the length of the shortest repeating sequence of digits is the length of the period of the decimal representation. For example, $61 / 495=0.1232323 \ldots$; although 23 and 2323 are sequences of digits that repeat in the decimal representation, the length of the shortest one is 2 and this is the length of the period.