Pure mathematics; including the higher parts of algebra and plane trigonometry, together with elementary spherical trigonometry Volume 2 - Softcover
Synopsis
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1875 edition. Excerpt: ...truth of the pi-oposition. 118. If &, b, c,...., 1 are a series of numbers prime to each other, and £ (n) represents how many numbers there uro which are prime to n and less than it, then / (abc 1) = 4 (a). f (b). £ (c) £ (1). First, suppose N = ab. We may arrange the numbers from 1 to ab in b columns thus: 1, 2, 3,... k,...a, es + 1, a+ 2, a+3,... a + k,...2a, 2a+1, 2a + 2, 2a + 3,... 2a + k,...3a, (6-1) a+l, (6-l)a + 2, (6-l)a + 3 (6-l)a + k,...a5. If we consider any column as the one commencing with k, we see that all the numbers in it are prime to a, or all the numbers contain some factor common with a, according as k is prime to a or not. Hence the number of columns which contain all the numbers of the column prime to a is equal to the number of quantities in the series 1, 2, 3,... k,... a, which are prime to a. And no other column contains any number prime to a. Hence j (a) is the number of columns containing numbers prime to a out of all the numbers from 1 to ab (1). Again, by Art. 114, Cor. 1, if the numbers in any column, as that commencing with k, be divided by b, the remainders, when k is prime to b, will form the series 1,2, 3,...., (5-1). Hence in every column there are as many numbers prime to 6 as there are of such numbers in the series 1, 2, 3,.... (6-1) Therefore f (b) is the number of quantities in each column which are prime to 6 (2). Hence from (1) and (2) we conclude that--The number of quantities in all the columns which are prime to a and prime to b, and therefore prime to ab, is 4, (a). f (b). Therefore f (ab) =j(a).j (b). If N = abc... J, we have j (abc I) = 4(a). l(bc T) = f (a). f (b). j (c... I) = &c. = f(a). j(b).f(c)....j(t). Q.E.D. Note. It will be seen that we consider here unity as...
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