Rational Numbers in Different Forms Practice
Math 8 Q1 (Pre-Algebra) / The Number System
How many digits are in the repeating cycle of 1727?(1 point)
$$
To determine how many digits are in the repeating cycle of a rational number expressed in decimal form, we can look at the remainder when the numerator is divided by the denominator.
In this case, we have the rational number $\frac{1727}{1}$. When we divide 1727 by 1, we get a quotient of 1727 and a remainder of 0. Since the remainder is 0, there is no repeating cycle. Therefore, there are 0 digits in the repeating cycle of 1727.
Answer: \boxed{0}.
wrong
incorrect
Apologies for the incorrect response. Let's try solving it correctly.
To find the repeating cycle in the decimal representation of a fraction, we can perform long division.
Let's divide 1727 by 1:
1727
_____________
1 | 1727
We see that 1 is a factor of 1727. So, the decimal representation of $\frac{1727}{1}$ is 1727.
Since 1727 has no decimal part, there is no repeating cycle.
Therefore, the number of digits in the repeating cycle of 1727 is \boxed{0}.
How many digits in 17/27
To find the number of digits in the repeating cycle of 1727, we need to convert it to a fraction and then look at the denominator. In this case, 1727 can be written as $$\frac{1727}{1}$$.
To convert this decimal to a fraction, we can use the fact that the decimal expands into a repeating cycle after the decimal point.
Let's start by multiplying both the numerator and denominator by 10 to move the decimal point one place to the right.
$$\frac{1727}{1} \times \frac{10}{10} = \frac{17270}{10}$$
Now, let's divide the numerator by the denominator to get the fraction in decimal form:
$$\frac{17270}{10} = 1727$$
Since the decimal part of the fraction repeats after the decimal point, we can conclude that the fraction $$\frac{1727}{1}$$ is equivalent to the decimal 1727 with a repeating cycle of 0. Therefore, the number of digits in the repeating cycle of 1727 is 0.
To find the number of digits in the repeating cycle of a repeating decimal, we need to convert the decimal into a fraction and then check the period of the fraction's decimal representation.
Step 1: Convert the repeating decimal into a fraction.
Let x be the repeating decimal 0.172727...
We can multiply both sides of the equation by a power of 10 to eliminate the repeating part:
1000x = 1727.272727...
10x = 17.272727...
Now we can subtract 10x from 1000x to eliminate the repeating part:
(1000x - 10x) = (1727.272727...) - (17.272727...)
990x = 1710
x = 1710/990
Step 2: Simplify the fraction.
The fraction 1710/990 can be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD):
GCD(1710, 990) = 90
1710/990 = 19/11
Step 3: Find the period of the fraction.
To find the period of the fraction's decimal representation, we can use the fact that the period of a fraction is equal to the smallest power of 10 that is congruent to 1 modulo the denominator.
In this case, the denominator is 11. We need to find the smallest power of 10 that satisfies 10^k ≡ 1 (mod 11).
We can check each power of 10 modulo 11:
10^1 ≡ 10 (mod 11)
10^2 ≡ 1 (mod 11)
Since 10^2 ≡ 1 (mod 11), the period of the fraction 19/11 is 2.
Step 4: Count the digits in the repeating cycle.
The repeating cycle of the decimal representation of 19/11 is 72.
Therefore, there are 2 digits in the repeating cycle of 1727.