1878.] Mr Glaisher, On circulating decimals. 189 



q" will contain (g'-l)*?*"^ digits, and ^ {q) = [q-V}q^\ Also 

 3* {k > 1) will have 6 periods and 487* will have 487 periods. 



(xiii) Unless g be a prime or a power of a prime, it must have 

 more than one period ; for there will be a single period only in the 

 case in which the period contains {q) digits. Now if q =r^sH"* ... 

 the number of figures in the period is the least common multiple 

 of pr*~\ ad'^, Tr~\ . . . ; and this cannot be equal to </) [q] unless 

 p = r — 1, a- = s - 1, &c. and r"^^ (r - 1), s^~^ (s - 1), ... be prime to 

 one another. But this cannot happen for r, s, ... being primes, 

 r — 1, s — 1, ... must have the common factor 2; and therefore 

 every number other than a prime or the power of a prime must 

 have at least two periods. 



§ 3. If q be not prime to 10, then q is either of the form 

 2'"5" or of the form 2'"5"5, where s is prime to 10. In the former 



case the fraction - is equal to a terminating decimal and the 



number of decimal places is equal to. r, where r is the greater of 

 the numbers m and n. 



If o' = 2'^5"5, then the decimal fraction equal to ■ „„ will 



2 5 5 



consist of r non-circulating digits (r being the greater of the 

 numbers m and ?i) followed by one of the periods of s as circu- 

 lating period. For, 



2"'5^s "" 10'' t s 



Now let the quantity in brackets =M+~, where M is an 



s 



integer and p'<S; then the given fraction 



lO*" "^ lO"- s ' 



that is to say the first r digits of the decimal are not periodic and 

 are obtained by dividing 2'""'5'"~"j; by s, and then one of the periods 

 of s commences. 



=•0931 122448979591836734693 

 877551020408163265306. 



