If (a,b)ε(0,1), then:
a/b
= _________________
0.a1a2.........anb1b2...........bm
Preperiod Period
Determine the respective lengths of the preperiod and the postperiod of (10!)
-1
*** Computer program/excel solver methodologies are welcome, but a semi-analytic (hand calculator and p&p) methodology is preferred.
I'm quite surprised that this problem has been out barely a day and we already have a "plug it in the calculator" solution as the official solution.
For a more analytical approach. we'll start with the prime factorization of 10! = 36628800 = (2^8*5*2) * (3^4*7).
The exponents of 2 and 5 are 8 and 2, respectively. The preperiod is the larger of these two exponents. Thus the length of the preperiod of (10!)^-1 is 8.
The the period of (10!)^-1 has the same period as 1/(3^4*7). 1/(3^4*7) can be expressed as a fraction with a denominator of all 9's. Then there are some x and y such that 1/(3^4*7) = x/(10^y - 1). In this arrangement, the smallest y is the length of the period, so we need to an integer y such that (10^y - 1) mod (3^4*7) = 0.
1/7 already is known to have a period of 6, so the valid choices for y start with 6,12,18,... Testing these finds y=18 gives an integer result for x. Thus the length of the period of (10!)^-1 is 18.
Solution
