What is the 43rd digit after the decimal point in the decimal representation of $\frac{1}{13}$?
The decimal representation of $\frac{1}{13}$ is $0.\overline{076923}$. Since the first six digits repeat, we know that after every 6th digit the pattern will restart. Since $43\div 6 = 7 r 1$, the first 42 digits will be seven repetitions of the same pattern followed by the first digit of the pattern. Since the first digit is $\boxed{0}$, this is our final answer.
0
Level 3
Number Theory
Problem: What is the 43rd digit after the decimal point in the decimal representation of $\frac{1}{13}$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the 43rd digit after the decimal point in the decimal representation of $\frac{1}{13}$, we first need to determine the decimal expansion of $\frac{1}{13}$. 

Calculating $\frac{1}{13}$ using long division, we find that it results in a repeating decimal: $0.\overline{076923}$. The repeating block "076923" has a length of 6 digits.

Next, we need to identify the position of the 43rd digit within this repeating sequence. To do this, we calculate the remainder of 43 when divided by 6, since the decimal repeats every 6 digits. 

Calculating $43 \mod 6$ gives us:
43 divided by 6 equals 7 with a remainder of 1. 

This means that the 43rd digit corresponds to the 1st digit in the repeating block "076923". 

Looking at the repeating block, the 1st digit is "0". 

Thus, the final answer is $0$.<|END_OF_LATENT|> The final answer is $0$.