208 V.R. G. T. BENIirP]TT ON THE RESIDUES OF POWERS OF NmiBERS 
The value of 
<f) (m) is 2 '^{p — {p — 1 ) (f “ h • . 
and, since p — \, q — I, . . . are all even, the L.C.M. can only equal (f) {m) when 
(i.) there is the only one odd prime 7 :) present; and 
(li.) /c = 1 or 0 . 
Hence the only moduli which admit of primitive roots (which exist only when the 
highest exponent is equal to (f> (pn)), are powers of odd primes, and double the powers 
of odd. primes. 
Examples .—What is the exponent of 3 for mod 1000000 = 2 ®. 5® ? 
Therefore 
The exp of 3 for mod 2 ® is 2 h 
,, „ 5® is 4 . 5®. 
exp of 3 mod 10® = 2'^’ , 5® = 50000. 
How many decimal places are there in the period of the product of ’Ol and •Ol, he., 
what is the exponent of 10 for mod 99 X 99 = 3“^. 11^ ? 
Exp of 10 for mod 3 = 1 . 
03 — 1 
5, ,, O i . 
3®= 3. 
04, 
11= 2. 
113 .j.:> 
) ) L ± ^ 
Therefore 
exp of 10 for mod 3^. iH = 9 X 22 = 198. 
Hence there are 198 figures in the period of (’01)^ 
How many decimals are there in the period of (•OOl)", he., what is the exponent 
of 10 for mod (999)“ = 37 \ 3 ^^? 
The exp of 10 mod 3 is 1 , 
and 
10 — 1 is divisible by 3®, 
therefore 
exp of LO mod 3®" is 3®" 
