10/17=0.5882352941176470 In the infinitely repeating decimal above ,five is the first digit in the repeating pattern ,what is the 264th digit? E. 1 F. 4 G. 5 H. 9
We can begin by multiplying both sides of the equation $10x=17.88235294117647$ by 10 to get $100x=178.8235294117647$. Then, we can subtract $10x=17.88235294117647$ from $100x=178.8235294117647$ to get $90x=160.94117647058823$. Dividing both sides by 90 gives $x=1.787679606228758$
We know that the repeating block starts with 5, and has length 7, so it must be $0.5882355\overline{882}$.
To find the 264th digit, we can note that the first digit of the repeating block (5) appears in the $1$st, $8$th, $15$th, $22$nd, $\ldots$ places. Continuing in this fashion, we can determine that the 264th digit is the same as the $264-7\cdot 37=5$th digit in the repeating block. Hence, the answer is $\boxed{\textbf{(G) }5}$.
To determine the 264th digit in the infinitely repeating decimal pattern, we need to find the pattern length first.
To do that, we can continue the division process until the pattern repeats. Let's calculate it step by step:
1) Divide 10 by 17:
10 ÷ 17 = 0 remainder 10
2) Multiply the remainder by 10 and divide it by 17 again:
(10 × 10) ÷ 17 = 100 ÷ 17 = 5 remainder 15
3) Repeat the process:
(15 × 10) ÷ 17 = 150 ÷ 17 = 8 remainder 14
4) Continue:
(14 × 10) ÷ 17 = 140 ÷ 17 = 8 remainder 4
5) Again:
(4 × 10) ÷ 17 = 40 ÷ 17 = 2 remainder 6
6) And again:
(6 × 10) ÷ 17 = 60 ÷ 17 = 3 remainder 9
7) Once more:
(9 × 10) ÷ 17 = 90 ÷ 17 = 5 remainder 5
The pattern repeats after 6 digits: 15 8 4 6 9 5.
Now let's find the 264th digit in this pattern:
- First, we divide 264 by 6 (the length of the pattern):
264 ÷ 6 = 44 remainder 0
Since there is no remainder, it means that the 264th digit is the last digit of the pattern, which is 5.
Therefore, the correct answer is G. 5.