How many digits are in the repeating cycle of 1727?(1 point)
To determine the repeating cycle of a number, we need to find the remainder when dividing 1 by the number. Let's do the long division process to find the repeating cycle of 1727:
```
0. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 7 2 7 0
-1 7 2 7
0 0 7 2
```
After the first division, we get a remainder of 072. Continuing the process, we have:
```
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 7 2 7 0
-1 7 2 7
0 0 7 2
- 1 7 2
2 5 0
- 2 4 2
6 0
```
Since we obtained the same remainder 072 after the second division, we can conclude that the repeating cycle of 1727 is 072. Therefore, the repeating cycle of 1727 has 3 digits. Answer: \boxed{3}.
How many digits are in the repeating cycle of 17/27?(1 point)
To find the repeating cycle of 17/27, we perform long division as follows:
```
0. 6 3 0 6 3 0 6 3 0 6 3 0 6 3 0 6 3 0 6 3
27 | 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 8
-1 7
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
1 0
- 9
```
As we can see, the remainder of each division is 10. Hence, the repeating cycle of 17/27 is 10. Therefore, the repeating cycle of 17/27 has 2 digits. Answer: \boxed{2}.
How many digits are in the repeating cycle of 1727?(1 point)
To find the number of digits in the repeating cycle of 1727, we need to divide 1 by 1727. Let's perform this division step-by-step:
1 ÷ 1727 = 0.00057965853812356979...
Since the decimal is non-repeating and not terminating, we need to find the repeating cycle.
To do that, we can observe the remainders obtained during the division.
When we divide 1 by 1727, the first remainder is 1.
To find the length of the repeating cycle, we need to continue the division process until we get a repeated remainder.
Continuing the division process, we get:
10 ÷ 1727 = 0.0057932816537467707355...
100 ÷ 1727 = 0.057932816537467707352...
1000 ÷ 1727 = 0.579328165374677073523...
.
.
.
As we can see, the remainder starts repeating after a few divisions. Hence, the repeating cycle of 1727 has 6 digits: 0, 0, 5, 7, 9, and 3.
Therefore, the repeating cycle of 1727 consists of 6 digits.
To find the number of digits in the repeating cycle of 1727, we need to understand its properties.
The repeating cycle of a decimal number occurs when the fraction representation of that number is non-terminating.
To find the repeating cycle, we can convert 1727 into a fraction. Since 1727 does not have any decimal places, we can write it as 1727/1.
To convert this fraction into a decimal, divide the numerator (1727) by the denominator (1). The result is 1727.
Since there are no decimal places and the fraction is not terminating, the repeating cycle in this case is zero.
Therefore, there are no digits in the repeating cycle of 1727.