Which of these fractions has the LONGEST repeating block in its decimal expansion?

Divide each one out. 1/11 = 0.090909… repeats a block of 2 digits; 1/37 = 0.027027… repeats a block of 3; 1/41 = 0.0243902439… repeats a block of 5; and 1/13 = 0.076923076923… repeats a block of 6. The longest belongs to 1/13. The size of the denominator is no guide at all — 41 is the biggest denominator here yet its block is shorter. What decides the length is the smallest power of 10 that leaves remainder 1 on division by the denominator: $10^2-1=99$ is a multiple of 11, while nothing shorter than $10^6-1$ works for 13.