How many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7?
1. Up to seven digits long
2. Up to six digits long
3. Infinitely long
4. Any finite length
The repeating cycle of a repeating decimal when the denominator is 7 will always have a maximum length of 6 digits.
Hence, the correct answer is option 2: Up to six digits long.
The correct answer is 2. Up to six digits long.
When the denominator of a rational number is 7, the repeating cycle of the repeating decimal can have a maximum length of 6 digits. This is because the repeating cycle is determined by the remainder when dividing the numerator by the denominator. Since there are only 6 possible remainders when dividing by 7 (0, 1, 2, 3, 4, 5), the repeating cycle can only have a maximum length of 6 digits. Any repeating cycle longer than 6 digits would imply a different denominator.
To determine the length of the repeating cycle of a repeating decimal, we can use the concept of remainders when dividing the numerator by the denominator.
In this case, the denominator is 7. When we divide the numerator by 7, the remainder could be any number in the range 0 to 6.
We know that the repeating cycle will continue until we encounter a remainder that we have encountered before. This is because once we encounter a repetitive remainder, the division process will repeat the same sequence of decimals.
Now, let's consider the remainders when dividing 1 by 7:
1 ÷ 7 = 0 remainder 1
10 ÷ 7 = 1 remainder 3
30 ÷ 7 = 4 remainder 2
20 ÷ 7 = 2 remainder 6
60 ÷ 7 = 8 remainder 4
40 ÷ 7 = 5 remainder 5
After 6 divisions, we can see that we have encountered remainder 1 again. This means that the repeating cycle of the decimal is 6 digits long. Therefore, the correct answer is:
Option 2: Up to six digits long.