IN THE THEORY OF NUMBERS. 409 



hence, (1-Pi«) 0-~Pi 2a ) ■ • • (1— i?/™ -1 ^") will be the same as 



Q-Pi*)&--Pi*') ■ ■ • (l-Pi Sm - 1 )> 

 where 5 X , d 2 , &c. are the remainders. But, by Prop. III., 8 1 8 r &c, are all different, 

 and are the numbers 1, 2, . . . m-1 : hence, 



(I-pKI-p 2 ) . . . (i-p TO - 1 )=(i-j Pl -)(i-i» 1 2 *) • • • 



=(i- Pl )(i- jPl 2 ). . .(l-ft— 2 ) 



Problem 2. If m be not a prime number, the same equation is true, by Prop. VII., 

 for p=p 1 ", 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 (l-p/) (l-pf*) . . . (l-jD 1 ( m - 1 >) 

 will be of the form l—p 1 y m =0; and, consequently, the product itself will be 

 equal to 0. 



