To find the number of digits in the repeating cycle of the fraction \( \frac{17}{27} \), we can first perform the long division to see how the decimal expansion behaves.
Dividing 17 by 27:
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17 divided by 27 gives 0 with a remainder of 17. So we can write: \[ 0. \text{ (we'll be repeatedly calculating the decimal)} \]
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Now we consider \( 170 \) (17 followed by a zero).
- \( 170 \div 27 \) gives 6 with a remainder of 8 (since \( 6 \times 27 = 162 \)).
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Next, we can look at \( 80 \) (the remainder):
- \( 80 \div 27 \) gives 2 with a remainder of 26 (since \( 2 \times 27 = 54 \)).
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Now consider \( 260 \) (the remainder):
- \( 260 \div 27 \) gives 9 with a remainder of 17 (since \( 9 \times 27 = 243 \)).
At this point, we notice that we have returned to the remainder of \( 17 \), which means we will start the cycle over again.
The decimal expansion is: \[ 0.629629629... \]
This means the repeating cycle is \( 629 \), which has 3 digits.
Thus, the number of digits in the repeating cycle of \( \frac{17}{27} \) is 3.
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