Find the smallest base ten positive integer with period 12.

Note: The period of an integer is the length of the repeating pattern of reciprocal. For example, repeating pattern of the reciprocal of 7 is 1/7 = 0.142857142857..... having a length of 6. So the period of 7 is 6.

A003060 in Sloane gives: 707, as the smallest number with reciprocal of length exactly 12. The cited text has the comment:

For n>0, a(n) is the least divisor d>1 of 10^n-1 such that the multiplicative order of 10 mod d is n.

(707 divides 999,999,999,999 exactly 1,414,427,157 times with no remainder.)

Compare e.g. http://perplexus.info/show.php?pid=7570&op=sol, a puzzle dealing with multiplicative orders.

 

 

Edited on November 30, 2012, 11:49 am

Posted by broll on 2012-11-30 11:33:19