To determine the number of digits in the repeating cycle of the fraction \( \frac{17}{27} \), we can first perform the division of 17 by 27.
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When dividing 17 by 27, you start with 0. The integer part is 0, and you can write it as \( 0.\) and then proceed with the division.
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Multiply \( 17 \) by \( 10 \) to get \( 170 \), and divide \( 170 \) by \( 27 \):
- \( 27 \) goes into \( 170 \) six times (since \( 27 \times 6 = 162 \)).
- Subtract \( 162 \) from \( 170 \) to get a remainder of \( 8 \).
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Multiply \( 8 \) by \( 10 \) to get \( 80 \), and divide \( 80 \) by \( 27 \):
- \( 27 \) goes into \( 80 \) two times (since \( 27 \times 2 = 54 \)).
- Subtract \( 54 \) from \( 80 \) to get a remainder of \( 26 \).
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Multiply \( 26 \) by \( 10 \) to get \( 260 \), and divide \( 260 \) by \( 27 \):
- \( 27 \) goes into \( 260 \) nine times (since \( 27 \times 9 = 243 \)).
- Subtract \( 243 \) from \( 260 \) to get a remainder of \( 17 \).
At this point, we see that we've returned to our original remainder of \( 17 \), indicating that the cycle will repeat from this point on.
The decimal representation of \( \frac{17}{27} \) is:
\[ \frac{17}{27} = 0.\overline{629} \]
The repeating part "629" has 3 digits. Therefore, the number of digits in the repeating cycle of \( \frac{17}{27} \) is:
\[ \boxed{3} \]