Question

how many digits long could the repeating cycle of a repeating deciaml be if the denomiator of the rational number is 7?

Answer 1

The length of the repeating cycle of a repeating decimal is determined by the denominator when the rational number is expressed in its simplest form. For a denominator of 7, you can find the length of the repeating cycle by determining the smallest integer \( n \) such that \( 10^n \equiv 1 \mod 7 \).

Here's how to calculate it:

1. **Calculate powers of 10 modulo 7**:
- \( 10^1 \mod 7 = 10 \mod 7 = 3 \)
- \( 10^2 \mod 7 = 10^2 \mod 7 = 100 \mod 7 = 2 \)
- \( 10^3 \mod 7 = 10^3 \mod 7 = 1000 \mod 7 = 6 \)
- \( 10^4 \mod 7 = 10^4 \mod 7 = 10000 \mod 7 = 4 \)
- \( 10^5 \mod 7 = 10^5 \mod 7 = 100000 \mod 7 = 5 \)
- \( 10^6 \mod 7 = 10^6 \mod 7 = 1000000 \mod 7 = 1 \)

2. **Find the smallest \( n \)**:
- We see that \( 10^6 \equiv 1 \mod 7 \), meaning the order of 10 modulo 7 is 6.

Therefore, the repeating cycle of the repeating decimal representation of a rational number with a denominator of 7 can be a maximum of **6 digits long**.