What is the 358th digit after the decimal point in the repeating decimal 0.317831?
Understand the Problem
The question is asking us to identify the 358th digit after the decimal point in the repeating decimal 0.317831. To solve this, we need to observe the repeating pattern and calculate which digit corresponds to the 358th position.
Answer
The 358th digit after the decimal point is $8$.
Answer for screen readers
The 358th digit after the decimal point in the repeating decimal $0.317831$ is 8.
Steps to Solve
- Identify the repeating pattern
The repeating decimal is $0.317831$, which has a repeating block of 6 digits: 317831.
- Determine the position in the pattern
To find the 358th digit, we need to find the position within the 6-digit repeating cycle. We calculate the position using modulo:
$$ 358 \mod 6 $$
- Calculate the modulo
Perform the calculation:
$$ 358 \div 6 = 59 \quad \text{(whole part)} \quad \text{and} \quad 358 - (59 \times 6) = 4 $$
So,
$$ 358 \mod 6 = 4 $$
- Find the corresponding digit
The remainder tells us the position within the repeating block.
- The digits in the block are:
- 1st: 3
- 2nd: 1
- 3rd: 7
- 4th: 8
- 5th: 3
- 6th: 1
Since the remainder is 4, the 4th digit in the repeating block is 8.
The 358th digit after the decimal point in the repeating decimal $0.317831$ is 8.
More Information
The repeating decimal consists of a sequence of 6 digits: 317831. Understanding how to break down repeating decimals and using modulus can simplify the extraction of specific digits.
Tips
- Confusing the repeating pattern length or miscalculating the modulo operation could lead to the wrong digit. Always double-check your calculations to ensure the correct digit is identified.
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