remaining factors of the proposed expressions. The last statement can be verified by trial, but when the student is acquainted with the subject of the resolution of algebraical expressions into factors it will be obvious on inspection. The resolution of algebraical expressions into factors is discussed in the Theory of Equations.

119. Next suppose we require the G. C. M. of three algebraical expressions A, B, C. Find the G. C. M. of two of them, say A and B; let D denote this G. C. M.; then the G. C. M. of D and C is the required G. C. M. of A, B and C.

For by Art. 111 every measure of D and C is a measure of A, B and C; and also every measure of A, B and C is a measure of D and C. Thus the G. C. M. of D and C is the G. C. M. of A, B and C.

120. In a similar manner we may find the G. C. M. of four algebraical expressions. Or we may find the G. C. M. of two of the given expressions and also the G. C. M. of the other two; then the G. C. M. of the two expressions thus found will be the G. C. M. of the four given expressions.

121. The definition and operations of the preceding articles of this chapter relate to polynomial expressions. The meaning of the term greatest common measure in the case of simple expressions will be seen from the following example:

Required the G. C. M. of 432a b'xy, 270a b3xz and 90a3bx3.

We find by Arithmetic the G. C. M. of the numerical coefficients 432, 270, and 90; it is 18. After this number we write every letter which is common to the simple expressions, and we give to each letter respectively the least index which it has in the simple expressions. Thus we obtain 18a bx, which will divide all the given simple expressions, and is called their greatest com

mon measure.

EXAMPLES OF THE GREATEST COMMON MEASURE.

Find the G. C. M. in the following examples:

5. ... 6x3-7ax2 - 20a3x and 3x2 + ax 4a2.

3x3-13x2+23x-21 and 6x3 + x2-44x + 21.

8. ... x1-3x2 + 2x2 + x −1 and x3- x2 - 2x + 2.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

... x*-7x3 + 8x2 + 28x-48 and x3- 8x2 + 19x - 14.

4x+9x3 + 2x2 2x-4 and 3x2+5x2 - x + 2.

2x-12x+19x2-6x+9 and 4x2-18x2 + 19x - 3.

6x+x3-x and 4x3- 6x2 - 4x + 3.

x3+ ax3-axy-y3 and x*+ 2x3y - a2x2+x2y2-2axy' — y*. 2x-11x-9 and 4x5 + 11x+81.

2a+3a3x-9ax and 6ax - 17a3x2 + 14a2x2 - 3ax1.

2x3 + (2α − 9) x3 — (9a + 6) x + 27 and 2x2 - 13x+18.

a3x3-a3bx2y+ab xy-b3y and 2a bx'y-ab'xy - b3y3. 12x-15yx + 3y and 6x-6yx + 2y3x-2y".

... x+3x-8x2 - 9x-3 and x5 - 2x2-6x3+ 4x2+13x + 6.

...

...

6x-4x-11x3-3x2-3x-1 and 4x2+ 2x3-18x2+3x-5.

x-ax3-a2x2-a3x - 2a and 3x3- 7ax +3a2x - 2a3.

VII. LEAST COMMON MULTIPLE.

122. In Arithmetic the least common multiple of two or more whole numbers is the least number which contains each of them exactly. The term is also used in Algebra, and its meaning in this subject will be understood from the following definition of the least common multiple of two or more Algebraical expressions. Let two or more Algebraical expressions be arranged according to descending powers of some common letter; then the expression of lowest dimensions in that letter which is divisible by each of these expressions is their least common multiple.

123. The letters L. C. M. will often be used for shortness instead of the term least common multiple; the term itself is not very appropriate for the reason already given in Art. 106.

Any expression which is divisible by another may be said t be a multiple of it.

124. We shall now shew how to find the L. C. M. of two Algebraical expressions. Let A and B denote the two expressions, and D their greatest common measure. Suppose A=aD and BbD. Then from the nature of the greatest common measure, a and b have no common factor, and therefore their least common multiple is ab. Hence the expression of lowest dimensions which is divisible by aD and bD is abD.

Hence we have the following rule for finding the L. C. M. of two Algebraical expressions: find their G. C. M.; divide either expression by this G. C. M., and multiply the quotient by the other expression. Or thus:-divide the product of the expressions by their G. C. M.

vious that every multiple of M is a common multiple of A d B.

126. Every common multiple of two Algebraical expressions is nultiple of their least common multiple.

Let A and B denote the two expressions, M their L. C. M.; let N denote any other common multiple. Suppose, if ssible, that when N is divided by M there is a remainder R; q denote the quotient. Thus R=N-qM. Now A and B asure M and N, and therefore (Art. 109) they measure R. t R is of lower dimensions than M; thus there is a common ltiple of A and B of lower dimensions than their L. C. M. This bsurd; hence there can be no remainder R; that is, N is a ltiple of M.

127. Next suppose we require the L. C. M. of three Algebraical ressions A, B, C. Find the L. C. M. of two of them, say A and let M denote this L. C. M.; then the L. C. M. of M and C is required L. C. M. of A, B and C.

For every common multiple of M and C is a common multiple 4, B and C (Art. 125). And every common multiple of A B is a multiple of M (Art. 126); thus every common multiof A, B and C is a common multiple of M and C. Therefore L. C. M. of M and C is the L. C. M. of A, B and C.

128. By resolving Algebraical expressions into their compofactors, we may sometimes facilitate the process of determintheir G. C. M. or L. C. M. For example, required the L. C. M. of a2 and x3- a3. Since

x2 — a2 = (x − a) (x+a) and x3 − a3 = (x − a) (x2 + ax + a2),

infer that x-a is the G. C. M. of the two expressions; conently their LCM is (x + a) (x3 — a3) that is.

129. The preceding articles of this Chapter relate to polynomial expressions. The meaning of the term least common multiple in the case of simple expressions will be seen from the following example. Required the L. C. M. of 432a b'xy, 270ab3x3z and 90a3bx3. We find by Arithmetic the L. C. M. of the numerical coefficients 432, 270 and 90; it is 2160. After this number we write every letter which occurs in the simple expressions, and we give to each letter respectively the greatest index which it has in the simple expressions. Thus we obtain 2160a*b3x3yz, which is divisible by all the given simple expressions, and is called their least common multiple.

130. The theories of the greatest common measure and of the least common multiple are not necessary for the subsequent Chapters of the present work, and any difficulties which the student may find in them may be postponed until he has read the Theory of Equations. The examples however attached to the preceding Chapter and to the present Chapter should be carefully worked, on account of the exercise which they afford in all the fundamental processes of Algebra.

EXAMPLES OF THE LEAST COMMON MULTIPLE.

1. Find the L. C. M. of 6x2-x-1 and 2x2 + 3x-2.

7. Find the L. C. M. of 2x-1, 4x2-1 and 4x2 + 1.

8. Find the L. C. M. of x3 — x, x3 − 1 and x3 + 1.