To determine the number of digits in the repeating cycle of the fraction \( \frac{17}{27} \), we can perform long division or recognize properties of fractions.
- To find the decimal representation, we compute \( \frac{17}{27} \):
- \( 17 \div 27 = 0.629629629... \)
From the division, we see that the decimal repeats with the digits "629".
- To confirm the length of the repeating cycle:
- The denominator \( 27 \) factors as \( 3^3 \). According to the properties of repeating decimals, the repeating cycle relates to the order of 10 modulo \( 27 \):
- We need to determine the smallest integer \( n \) such that \( 10^n \equiv 1 ,(\text{mod}, 27) \).
Now we calculate:
- \( 10^1 \mod 27 = 10 \)
- \( 10^2 \mod 27 = 100 \mod 27 = 19 \)
- \( 10^3 \mod 27 = 1000 \mod 27 = 1 \)
The smallest \( n \) satisfying this is \( n = 3 \).
Thus, the length of the repeating cycle of \( \frac{17}{27} \) is 3.
Final answer: 3 digits.
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