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.
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The attachment should enable you to give a proof:
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.
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