408 



BALFOUR STEWART, ESQ., ON A PROPOSITION 



1 to m— 1 inclusive, though not in order of magnitude. We have exhibited this 

 in the margin for m = 7, the multipliers being 3 and 4. 



IV. If m be not a prime, but composed of the prime factors axbxcx 

 dxexfx . . . . , and if a or one of its prime factors (/) be also one of the prime 

 factors of m, then in some case will a(3=ym, where (3 is one of the numbers 

 1, 2, 3 m^l. 



For -, and j are whole numbers < m ; and a . — = - . m, which satisfies the 



condition. 



V. If neither a nor any of its prime factors is also a prime factor of m, then 

 there will be a remainder 8 <m whatever be (3. 



For if not, let a(3=ym; then since my (3 .-. y<a. Therefore 7 is either 

 prime to a, or there is a factor in a which is not in 7 : but this factor is, by hy- 

 pothesis, not in m : it is, consequently, not in 7 m, which is absurd. 



VI. The remainders will be different for every different value of (3. For 

 if possible, let a /3 = 7 m + 8 



a (/3 + {3 1 ) = y l m + 8 . 



= (7 t — 7) m + a (3 

 therefore, a j3 l = (y 1 — 7) m which, as in the last Prop., is impossible. 



VII. If, then, we arrange the numbers as in Prop. III., 

 we shall have amongst the remainders, for all values 

 of a which either divide m, or have a prime factor in com- 

 mon with it ; whilst, for all other values of «, we shall 

 have the same results as in Prop. III. This is shown for 

 m=8 in the margin, for the multipliers 3, 4, and 6. 



1 



2 



3 



4 



5 



6 



7 



2 















3 



6 



1 



4 



7 



2 



5 



4 







4 







4 







4 



5 















6 



4 



2 







6 



4 



2 



7 















Problem 1. If m be a prime number, the roots of the equation a™ -1=0 are 



pm — 1 where ». = cos -— + \/-l sin — 



1 mi. 



m 



For, from the theory of equations, 



or 



putting 1 for x, 

 If p= Pl «, 



m=(l- Pl )(l- Pl 2) . . . (l- Pl m - l ). 

 since Pl m =l, p/ m =l; 



and if a (3=ym = 8> 



Pi a/3 =i>i 2 



