Pairwise distinct primes

Aug 2013
2
0
Australia
Let n = p1.....pm for pairwise distinct primes pi other than 2 and 5. Let ki
be the period of the decimal expression of 1/pi. Show that the period of the decimal
expression of 1/n is the least common multiple of k1,....,km.
 
Dec 2012
1,145
502
Athens, OH, USA
Hi,
The attachment should enable you to give a proof:

MHFnumbertheory3.png
 
  • Like
Reactions: 2 people

emakarov

MHF Hall of Honor
Oct 2009
5,577
2,017
A proof of the first fact can be found in Elementary Number Theory by Dudley, which is available online.

Section 15, p. 118, Theorem 4. If (n, 10) = 1, then the period of 1/n is the order of 10 modulo n.
 
  • Like
Reactions: 1 person