The maximum number of digits in the repeating block of the decimal expansion of 1/7 is 6.
According to the reference provided, the decimal representation of 1/7 has a repeating block of digits. Specifically, the reference states: "Therefore, the maximum number of digits in the repeating block of digits in the decimal expansion of 1/7 is 6." This confirms that after the decimal point, the sequence of digits that repeats has a length of six.
Here's a breakdown of why this is the case:
- When you perform the long division of 1 by 7, you will notice a recurring pattern of remainders.
- Each remainder corresponds to a unique digit in the quotient (the decimal representation).
- The remainders cannot be more than one less than the divisor. In the case of dividing by 7, the possible remainders are 0, 1, 2, 3, 4, 5, and 6.
- A remainder of 0 means the division terminates, but in this case, the division doesn't terminate.
- Once a remainder repeats, the quotient digits also start to repeat.
- The repeating block will have a maximum of 6 digits in this case since there are a maximum of 6 non-zero remainders possible for a divisor of 7.
The decimal representation of 1/7 is 0.142857142857..., where the block '142857' repeats infinitely.
