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