114 History of the Theory of Numbers. [Chap, v 
[cf. Poinsot^^] to be divisible by r; after excluding them we get N"{r — l)/r 
numbers; etc. 
Euler^ noted in a posthumous paper that, if p, q, r are distinct primes, 
there are r multiples ^pqr of pq, and qr multiples of p, and a single multiple 
of pqr, whence 
<t){pqr)=pqr-qr-pr-pq-^r+p-\-q-l = {p-l){q-l){r-l). 
In general, if M is any number not divisible by the prime p, and if fx 
denotes the number of integers ^M and prime to M, there are M—fj. 
integers ^M and not prime to M and hence p''{M—ii) integers ^Mp" and 
not prime to M and therefore not prime to Mp". Of the Mp^~^ multiples 
^Mp^ of p, exclude the p"~^(ilf — /i) which are not prime to M; we obtain 
p^'V multiples of p which are prime to M. Hence 
<^(p"M)=p"M-p''(M-m)-p""V = P"~Hp-1)m- 
A. M. Legendre^ noted that, if ^, . . . , co are any odd primes not dividing 
A, the number of terms of the progression A+B, 2A+5, . . . , nA-\-B which 
are divisible by no one of the primes 0, . . . , co is approximately n(l — 1/0) . . . 
(1 — 1/co), and exactly that number if n is divisible by 0, . . . , co. 
C. F. Gauss^ introduced the symbol (i>{N). He expressed Euler's^ proof 
of (1) in a different form. Let a be any one of the 4>{A) integers <A and 
prime to A, while j8 is any one of the 4>{B) integers <B and prime to B. 
There is one and but one positive integer x<AB such that x=a (mod A), 
x=^ (mod B). Since this x is prime to A and to B, it is prime to AB. 
Making the agreement that <^(1) = 1, Gauss proved 
(4) 20(d) =N {d ranging over the divisors of N). 
For each d, multiply the integers ^d and prime to d by N/d; we obtain 
S0(d) integers ^N, proved to be distinct and to include 1, 2, . . ., iV. 
A. M. Legendre^ proved (3) as follows: First, let N = pM, where p is a 
prime which may or may not divide M; then Mp—M of the numbers 
1,. . ., N are not divisible by p. Second, let N = pqM, where p and q are 
distinct primes. Then 1,. . ., N include M numbers divisible by both p 
and q; Mp — M numbers divisible by q and not by p; Mq — M numbers 
divisible by p and not by q. Hence there remain A^(l — l/p)(l — 1/q) num- 
bers divisible by neither p nor q. Third, a like argument is said to apply 
to N = pqrM, etc. 
Legendre (p. 412) proved that ii A,C are relatively prune and if 0,X,ju, . . . , 
CO are odd primes not dividing A, the number of terms kA — C{k = l,...,n), 
which are divisible by no one of 0, . . . , to, is 
*Tractatu3 de numerorum, Comm. Arith., 2, 515-8. Opera postuma, I, 1862, 16-17. 
*Es3ai siir la thdorie des nombres, 1798, p. 14. 
'Disquisitionea Arithraeticse, 1801, Arts. 38, 39. 
•Th^orie des nombres, ed. 2, 1808, 7-8; German trans, of ed. 3 by Maser, 8-10. 
