IN THE THEORY OF NUMBERS. 409 
hence, (1—p,.) 1—p,?%) . . . (L—p,™-*) will be the same as 
| (l—p,”) dpi) . .. d—p"-) 
where 6,, 6,, &c. are the remainders. But, by Prop. III., 6, 6,, &c., are all different, 
and are the numbers 1, 2, . . . m—1: hence, 
(1—p) (1—p?) . . . (1—p™~1)=(1—p,*) (I—p,?*) - - - 
=(1—p,) (l—p,’) . . .(—p,"~) 
=m, 
Problem 2. If m be not a prime number, the same equation is true, by Prop. VII., 
_ for p=p,*, where a is less than, and prime to m; but if a or one of its prime factors 
is a prime factor of m, then one of the factors (1—p,”) (l—p,2%) . . . (—p,("™-") 
will be of the form 1—p,»”=0; and, consequently, the product itself will be 
equal to 0. 
