The Number That Cycles

When a fraction repeats, the length of its block is a number-theory puzzle. 1/7 = 0.142857142857..., and "142857" is one of math's most famous cyclic numbers: multiply it by 2, 3, 4, 5, or 6 and the same digits simply rotate into a new order.

For a prime denominator p, the repeating block can be at most p − 1 digits long. Primes that hit that maximum — like 7, whose block is 6 digits — are called full-reptend primes.