The decimal expansion of 1/271 repeats with a period of length 5:
.003690036900369 ...

However, it is not the smallest number q for which the decimal expansion of 1/q has a repetition length of 5.

Find the smallest q so that the decimal expansion of 1/q has repetition length n for each of {1, 2, ..., 10}

Is there a simple way of finding such a number?

(In reply to re(3): General solution by Old Original Oskar!)

But it must be the smallest such that meets these requirements you have specifiied -- i.e. (using Jer's notation) the smallest natural number q such that 1/q has repetition length n is the least divisor of 2^n-1 that does not divide any of the numbers 2^k-1 for 0<k<n.

Posted by Richard on 2006-09-26 23:07:48