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What can the maximum number of digits be in the repeating block of digits in the decimal expansion of ? Perform the division to check your answer.
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Q. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of ? Perform the division to check your answer.
- Perform Long Division: To determine the maximum number of digits in the repeating block of the decimal expansion of , we need to perform the long division of by .
- Determine Fits into : We start the division by determining how many times fits into (since does not fit into , we add a decimal point and a zero to , making it , and then another zero, making it ). goes into a total of times (), leaving a remainder of .
- Calculate Remainder : We bring down another zero to the remainder, making it , and divide again by . goes into a total of times (), leaving a remainder of .
- Divide by Again: We bring down another zero to the remainder, making it , and divide again by . goes into a total of times (), leaving a remainder of .
- Repeat Remainders: We bring down another zero to the remainder, making it , and divide again by . goes into a total of times (), leaving a remainder of .
- Identify Repeating Block: We bring down another zero to the remainder, making it , and divide again by . goes into a total of times (), leaving a remainder of .
- Identify Repeating Block: We bring down another zero to the remainder, making it , and divide again by . goes into a total of times (), leaving a remainder of .We bring down another zero to the remainder, making it , and divide again by . goes into a total of times (), leaving a remainder of .
- Identify Repeating Block: We bring down another zero to the remainder, making it , and divide again by . goes into a total of times (), leaving a remainder of .We bring down another zero to the remainder, making it , and divide again by . goes into a total of times (), leaving a remainder of .We notice that we have seen the remainder before, which means the digits will start to repeat from this point. The remainders we have seen so far are , , , , and , which correspond to the digits , , , , and in the decimal expansion.
- Identify Repeating Block: We bring down another zero to the remainder, making it , and divide again by . goes into a total of times (), leaving a remainder of .We bring down another zero to the remainder, making it , and divide again by . goes into a total of times (), leaving a remainder of .We notice that we have seen the remainder before, which means the digits will start to repeat from this point. The remainders we have seen so far are , , , , and , which correspond to the digits , , , , and in the decimal expansion.Since the remainders are starting to repeat, we have found the repeating block of the decimal expansion of . The repeating block is , which has digits.
