of C and D. Therefore every common measure of A and B is a measure of D. But no expression of higher dimensions than D can divide D. Therefore D is the greatest common measure of A and B.

108. It is obvious that, every measure of a common measure of two or more expressions is a common measure of those expressions.

109. It is shewn in Art. 107 that every common measure of A and B measures D; that is, every common measure of two expressions measures their greatest com

mon measure.

110. We shall now state and exemplify a rule which is adopted in order to avoid fractions in the quotient; by the use of the rule the work is simplified. We refer to the chapter on the Greatest Common Measure in the larger Algebra, for the demonstration of the rule.

Before placing a fresh term in any quotient, we may divide the divisor by any expression which has no factor which is common to the expressions whose greatest common measure is required; or, we may multiply the dividend at such a stage by any expression which has no factor that occurs in the divisor.

111. For example; required the G. C.M. of 2x2 −7x+5 and 3x2-7x+4. Here we take 2x2-7x+5 as divisor; but if we divide 3x2 by 2x2 the quotient is a fraction; to avoid this we multiply the dividend by 2, and then divide.

2x2-7x+5)6x2-14x+ 8(3

6x2-21x+15

7x-7

If we now make 7x-7 a divisor and 2x2-7x+5 the dividend, the first term of the quotient will be fractional; but the factor 7 occurs in every term of the proposed divisor, and we remove this, and then divide.

x−1) 2x2-7x+5 (2x−5

2x2-2x

−5+5

-5 +5

Thus we obtain x-1 as the G. C. M. required.

Here it will be seen that we used the second part of the rule of Art. 110, at the beginning of the process, and the first part of the rule later. The first part of the rule should be used if possible; and if not, the second part. We have used the word expression in stating the rule, but in the examples which the student will have to solve, the factors introduced or removed will be almost always numerical factors, as they are in the preceding example.

We will now give another example; required the G. C.M. of 2x4-7x34x2+x-4 and 3x4-11x3- 2x2-4x-16. Multiply the latter expression by 2 and then take it for dividend.

2x1 — 7x3 — 4x2 + x − 4) 6x1 − 22x3 – 4x2-8x-32(3

6x4-21x3-12x2+3x-12

We may multiply every term of this remainder by 1 before using it as a new divisor; that is, we may change the sign of every term.

x3-8x2+11x+20)2x1— 7x3- 4x2+x−4(2x+9

2x-16x3+22x2 + 40x

9x3-26x2- 39x-4

9x3-72x2+ 99x+180

46x2-138x-184

Here 46 is a factor of every term of the remainder; we remove it before using the remainder as a new divisor.

112. Suppose the original expressions to contain a common factor F, which is obvious on inspection; let A=aF and B=bF. Then, by Art. 109, F will be a factor of the G. C. M. Find the G. C. M. of a and b, and multiply it by F; the product will be the G. C. M. of A and B.

113. We now proceed to the G.C.M. of more than two compound expressions. Suppose we require the G.C. M. of three expressions A, B, C. Find the G.C. M. of any two of them, say of A and B; let D denote this G. C. M.; then the G.C. M. of D and C will be the required G. C. M. of A, B, and C.

For, by Art. 108, every common measure of D and C is a common measure of A, B, and C; and by Art. 109 every common measure of A, B, and C is a common measure of D and C. Therefore the G. C. M. of D and C is the G.c. M. of A, B, and C.

114. In a similar manner we may find the G. C. M. of four 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 results thus obtained will be the G.C.M. of the four given expressions.

EXAMPLES. XII.

Find the greatest common measure in the following examples.

1. 15, 18x2.

3. 36, 48x54.

5. 4(x+1), 6(x2-1).

2. 16a2b3, 20a3b2.
4. 35a2b3x3y*, 49a2bax1y3.
6. 6(x+1)3, 9(x2-1).

7. 12 (a+b), 8(a-b4). 8. x-y3, x^—y1.

9. x2+8x+15, x2+9x+20.

10. x2-9x+14, x2-11x+28.

11. x2+2x-120, x2-2x-80.

12. x2-15x+36, x2-9x-36.

13. x3+6x2+13x+12, x3+7x2+16x+16.

14.

x3-9x2+23x-12, x3-10x2+28x-15.

15. x3-29x+42, x3+x2-35x +49.

16. 3-41x-30, x3-11x2 + 25x+25.

17. x3+7x2+17x+15,

x3+8x2+19x+12.

18. x3-10x2+26x-8, x3 — 9x2+23x-12.

19. 4(x+1), 3(x+22+1).

20. 5(x2−x+1), 4(x −1).

21. 6x2+x-2, 9x3 +48x2 + 52x + 16.

22.

x3-4x2+2x+3, 2x1-9x3 + 12x2 +7.

23. x2+x2-6, x3-3x2+2.

24. x3-2x2+3x-6, x2-x3 — x2 — 2x.

-

25. x2-1, 3x+2x2 + 4x3 + 2x2+x.

26.

27.

x4-9x2-30x-25, x5+x1-7x2+5x.

35x3+47x2 + 13x+1, 42x+41x3-9x2-9x-1.

28. x6-3x+6xa— 7x3 +6x2-3x+1,

x6x5+ 2x4 — x3 + 2x2- x + 1.

29. 2x2-6x3+3x2 −3x+1, x2 -3x+x3-4x2 + 12x−4. 30. x3-1, x1o+x3 + x3 +2x2 + 2x1+2x3 + x2+x+1.

31.

2-3x-70, x3-39x+70, x3-48x+7.

32. x2-xy-12y2, x2+5xy+6y2.

33. 2x2+3ax+ a2, 3x2+2ax-a2.

34. x3-3a2x-2a3, x3-ax2 - 4a3.

35. 3x3-3x2y+xy2-y3, 4x2y— 5xy2+y3.

XIII. Least Common Multiple.

115. In Arithmetic a whole number which is measured by another whole number is said to be a multiple of it; a whole number which is measured by two or more whole numbers is said to be a common multiple of them.

116. In Arithmetic the least common multiple of two or more whole numbers is the least whole number which is measured by them all. The term least common multiple is also used in Algebra, but here it is not very appropriate; see Art. 98. The letters L. C. M. will often be used for shortness instead of this term.

We have now to explain in what sense the term is used in Algebra.

117. It is usual to say, that by the least common multiple of two or more simple expressions, is meant the least expression which is measured by them all; but this definition will not be fully understood until we have given and exemplified the rule for finding the least common multiple of simple expressions.

The following is the Rule for finding the L. C. M. of simple expressions. Find by Arithmetic the L. C. M. of the numerical coefficients; after this number put every letter which occurs in the expressions, and give to each letter respectively the greatest index which it has in the expressions.

118. For example; required the L. C. M. of 16abc_and 20a3b3d. Here the numerical coefficients are 16 and 20, and their L. C. M. is 80. The letters which occur in the expressions are a, b, c, and d; and their greatest indices are respectively 4, 3, 1, and 1. Thus we obtain 80a4b3cd as the required L. C. M.

Again; required the L. C. M. of 8a2b3c2.xyz3, 12abcx2y3, and 16a3exy. Here the L. C. M. of the numerical coefficients is 48. The letters which occur in the expressions are a, b, c, x, y, and z; and their greatest indices are respectively 4, 3, 3, 5, 4, and 3. Thus we obtain 48a4b3c3ay13 as the required L.C.M.