Mathematics Problem Of the Week

Spring 2004

POW 4

Prime Time

1). Show that for each prime number p (except p = 2 or p = 5) that there exists a number of the form n = 111···1 such that p divides n, (for example 3 divides 111 and 7 divides 111111). Hint: Fermat’s little theorem from number theory implies that for any number p ≠ 2 or 5

p divides .

2). Find the sum of the infinite series:

in which each term has no prime factor except 2 or 5. Hint: Recall from the theory of geometric series that

where a is the first term and r is the ratio between terms.

Due Friday, February 20^th at Noon.

Solutions should be submitted to Dr. Dan Gagliardi’s mailbox in the Math office or sent via e-mail to dgagliardi@stlawu.edu. Presentation counts! The prize-winning entry will be selected from all correct submissions, based on the clarity, creativity and elegance of the solution.

Look for the SLU POW on the Web at http://it.stlawu.edu/~math/activities/