Which digit will appear in the 534th place after the decimal point in the decimal representation of $\frac{5}{13}$?
The decimal representation of $\frac{5}{13}$ is $0.\overline{384615}$, which repeats every 6 digits. Since 534 is a multiple of 6, the 534th digit is the same as the last digit in the repeating block, which is $\boxed{5}$.
5
Level 3
Number Theory
Problem: Which digit will appear in the 534th place after the decimal point in the decimal representation of $\frac{5}{13}$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the digit in the 534th place after the decimal point in the decimal representation of 5/13, first, we need to find the decimal expansion of 5/13. Performing long division of 5 by 13 gives us the repeating decimal 0.384615, which has a period of 6 digits: 384615.

Next, we need to identify the position of the 534th digit within this repeating sequence. To do this, we calculate the remainder of 534 when divided by the length of the repeating sequence, which is 6. 

Calculating 534 mod 6:
534 ÷ 6 = 89 with a remainder of 0.

This means that the 534th digit corresponds to the last digit of the repeating sequence. The repeating sequence is 384615, and the last digit in this sequence is 5.

Thus, the digit that appears in the 534th place after the decimal point in the decimal representation of 5/13 is 5.<|END_OF_LATENT|> The final answer is $5$.