CHAPTER XL 
MISCELLANEOUS THEOREMS ON DIVISIBILITY, GREATEST 
COMMON DIVISOR. LEAST COMMON MULTIPLE. 
Theorems on Divisibility. 
An anonymous author^ noted that for n a prime the sum of 1 , 2, . . . , n — 1 
taken by twos (as 1+2, 1+3,. . .), by fours, by sixes, etc., when divided 
by n give equally often the residues 1, 2,..., n — 1, and once oftener the 
residue 0. The sum by threes, fives, . . . , give equally often the residues 
1,. . ., n — 1 and once fewer the residue 0. 
J. Dienger^ noted that if w''-+'±l and (m^'-+2_ 1)7(^2 _i) are divisible 
by the prime p, then the sum of any 2r+l consecutive terms of the set 
1, m^", m^'^", m^'^", . . . is divisible by p. The case m = 2, r = l, p = 3 or 7 
was noted by Stifel (Arith. Integra). 
G. L. Dirichlet^ proved that when n is divided by 1, 2, . . ., n in turn 
the number of cases in which the remainder is less than half the divisor 
bears to n a ratio which, as n increases, has the limit 2 — log 4 = 0.6137 
. . . ; the sum of the quotients of the n remainders by the corresponding 
divisors bears to n a ratio with the limit 0.423 .... 
Dirichlet^ generalized his preceding result. The number h of those 
divisors 1,2,. . . , p (p^ ti), which yield a remainder whose ratio to the divisor 
is less than a given proper fraction a, is 
-liH-B-"]} 
Assuming that pVn increases indefinitely with n, the limit of /i/p is a 
if n/p increases indefinitely with n, but if n/p remains finite is 
J. J. Sylvester^ noted that 2"""^^ is a factor of the integral part of /c^"*"*"^ 
and of the integer just exceeding h^"^, where ^ = l + \/3- 
N. V. Bougaief^ called a number primitive if divisible by no square >1, 
secondary if divisible by no cube. The number of primitive numbers ^ n is 
H,{n)=i:q{u)+iq{u)+.. ., <i = [VnA'], 
1 1 ' 
where q{u) is zero if u is not primitive, but is +1 or —1 for a primitive u, 
according as ?/ is a product of an even or odd number of prime factors. 
iJour. fiir Math., 6, 1830, 100-4. ^Archiv Math. Phys., 12, 1849, 425-9. 
3Abh. Ak. Wiss. BerMn, 1849, 75-6; Werke, 2, 57-58. Cf. Sylvester, Amer. Jour. Math., 5, 
1882, 298-303; CoU. Math. Papers, IV, 49-54. 
<Jour. fur. Math., 47, 1854, 151-4. Berlin Berichte, 1851, 20-25; Werke, 2, 97'-104; French 
transl. by O. Terquem, Nouv. Ann. Math., 13, 1854, 396. 
^uar. Joum. Math., 1, 1857, 185. Lady's and Gentleman's Diary, London, 1857, 60-1. 
"Comptes Rendus Paris, 74, 1872, 449-450. BuU. Sc. Math. Astr., 10, I, 1876, 24. Math. 
Sbornik (Math. Soc. Moscow), 6, 1872-3, I, 317-9, 323-331. 
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