What are two primes p > 0 for which 1/p has a purely periodic decimal expansion with a period 5 digits long?

Source: California math competition 2001

For the period to be 5 digits long, it has to divide 99999. The factorization of 99999 is 3^2*41*271. 1/3 has a period of 1, but 1/41 and 1/271 have a period of 5. Therefore, the two primes are 41 and 271.

 

Posted by Math Man on 2011-12-16 10:25:24