R', and therefore cannot be greater than R'. And, if R' exactly divides R, it will also divide B, since B=RQ'+R'; and whatever exactly divides B and R, will also exactly divide A, since A=BQ +R; therefore, if R' exactly divides R, it will exactly divide both A and B, and will be their greatest common divisor.

In the same manner, by continuing to divide the last divisor by the last remainder, it may always be shown, that the greatest common divisor of A and B will exactly divide every new remainder, and, of course, cannot be greater than either of them. It may also always be shown, as above, in the case of R', that any remainder, which exactly divides the preceding divisor, will also exactly divide A and B. Then, since the greatest common divisor of A and B cannot be greater than this remainder, and as this remainder is a common divisor of A and B, it will be their greatest common divisor sought.

The same principle may be illustrated by numbers, by calling A, 55, and B, 15, and proceeding to find their greatest common divisor.

ART. 102. When the remainders decrease to unity, or when we arrive at a remainder which does not contain the letter of arrangement, it is evident that there is no common divisor of the two quantities.

ART. 103. If either quantity contains a factor not found in the other, that factor may be canceled without affecting the common divisor. Thus, in the two quantities, x(x2—y2) and y(x2+2xy+y2), · of which the greatest common divisor is x+y, we may cancel x in the first, or y in the second, or both of them, and the greatest common divisor of the resulting quantities will still be x+y.

ART. 104. We may multiply either quantity by a factor not found in the other, without changing the greatest common divisor. Thus, in the two quantities, x(x2 y2) and y(x2+2xy+y2), if we multiply the first by m and the second by n, we have mx(x2—y2) and ny(x2+2xy+y2), of which the greatest common divisor is still x+y.

ART. 105. But if we multiply either quantity by a factor found in the other, we change the greatest common divisor. Thus, in the two quantities, x(x2—y2) and y(x2+2xy+y2), if we multiply the second by x, the two quantities become x(x2-y2) and xy(x2 +2xy+y2), of which the greatest common divisor is x(x+y), instead of x+y as before. In like manner, if we multiply the first quantity by y, the greatest common divisor of the two resulting quantities will be y(x+y).

ART. 106. From Art. 101 it is evident that the greatest common divisor of two quantities will exactly divide each of the successive remainders; therefore, the principles of the three preceding articles apply to the successive remainders that arise in finding the greatest common divisor.

ART. 107. It is evident that any common factor of two quantities, must also be a factor of their greatest common divisor. Where such common factor is easily seen, as when it is a monomial, it simplifies the operation to set it aside, and find the greatest common divisor of the remaining quantities.

We shall now show the application of these principles.

OPERATION.

x3-xz2
xz2-23

1. Find the greatest common divisor of x3-23 and x1—x222. Here the second quantity contains x2 as a factor, but it is not a factor of the first; we may therefore cancel it (Art. 103), and the second quantity becomes x2-22. Then divide the first by it. After dividing, we find that z2 is a factor of the remainder, but not of x2-z2, the next dividend. We therefore cancel it (Art. 103), and the second divisor becomes x-z. Then, dividing by this, we find there is no remainder; therefore x-z is the greatest common divisor.

2. Find the greatest common divisor of x5 +x21⁄23 and x3—x3z2.

The factor x2 is common to both quantities; it is, therefore, a factor of the greatest divisor (Art. 107), and may be taken out and reserved. Doing this, the quantities become x3+23 and x3-xz2. The second quantity still contains a common factor, x, which the first does not; canceling this, it becomes x2-2. Then proceeding as in the first example, we find that x+z divides without a remainder; therefore, x2(+2) is the required greatest common divisor.

or (x-2)z2

x2

x2

+%

OPERATION.

x3+z3 \x2-
x3-xx3 \x
xz2+23
or (x+2)z2

x2-z2 1x+z x2+xxx-x

3. Find the greatest common divisor of 10a2x2—4a2x—6a2, and 5bx2-11bx+6b.

By separating the monomial factors, we find

10a2x2-4a2x-6a2-2a2 (5x2-2x-3),

and 5bx2-11bx+6b÷b(5x2—11x+6).

[OVER.]

4. Find the greatest common divisor of 4a2-5ay+y2, and 3a3-3a2y+ay2—y3.

In solving this exam

ple, there are two in

stances in which it is ne

OPERATION.

3 a3—3a2y+ay2—y3|4a2—5ay+ y2

4

cessary to multiply the 12a3-12a1y+4ay2-4y3 3a+3y

ART. 108. From the preceding demonstrations and examples, we derive the following

RULE FOR FINDING THE GREATEST COMMON DIVISOR OF TWO POLYNOMIALS.-1. Divide the greater polynomial by the less, and if there is no remainder, the less quantity will be the divisor sought.

2. If there be a remainder, divide the first divisor by it, and continue to divide the last divisor by the last remainder, until a divisor is obtained which leaves no remainder; this will be the greatest common divisor of the two given polynomials.

NOTES.-1. When the highest power of the leading letter is the same in both, it is immaterial which of the quantities is made the dividend.

2. If both quantities contain a common factor, let it be set aside, as forming a factor of the common divisor, and proceed to find the greatest common divisor of the remaining factors, as in Example 2.

3. If either quantity contains a factor not found in the other, it may be canceled before commencing the operation, as in Example 3. See Art. 103.

4. Whenever it is necessary, the dividend may be multiplied by any quantity which will render the first term exactly divisible by the first term of the divisor. See Art. 104.

5. If, in any case, the remainder does not contain the leading letter, there is no common divisor.

6. To find the greatest common divisor of three or more quantities, first find the greatest common divisor of two of them; then of that divisor and one of the other quantities, and so on. The last divisor thus

found will be the greatest common divisor sought.

7. Since the greatest common divisor of two quantities contains all the factors common to both, it may be found most easily by separating the quantities into factors, where this can be done by the rules for factoring. Arts. 92 to 95.

Find the greatest common divisor of the quantities in each of the following

EXAMPLES.

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

2. 9x2-4 and 9x2-15x-14.

3. a2-ab-12b2 and a2+5ab+6b2.

4. 4a2-b2 and 4a2+2ab-2b2.

5. a1—x1 and a3—a2x—ax2—x3.

6. x3-5x2+13x-9 and x3-2x2+4x-3.

7. x3-5x2+16x-12 and x3-2x2-15x+16.

8. 21x3-26x2+8x and 6x2-x-2.

Ans.x-1. Ans. 3x+2.

Ans. a+3b.

Ans. 2a-b.

Ans. a2-x2.

Ans.x-1.
Ans.x-1.

Ans. 3x-2.

9. 2x+11x3-13x2-99x-45 and 2x3-7x2-46x-21.

Ans. 2x2+7x+3.

Ans. x2-2x+3.

Ans. 12x-5.

Ans. x+4. Ans. x2+ax+a2.

10. x42x2+9 and 7x3-11x2+15x+9. 11. 48x2+16x-15 and 24x3-22x2+17x-5. 12. x2+5x+4, x2+2x-8, and x2+7x+12. 13. x1+a2x2+aa and x2+ax3—a3x—aa. 14. 2b-10ab2+8a2b and 9a1-3 ab3+3a2b2—9a3b. Ans. a--b. 15. x-px3(9-1)x2+px-q and x-qx3-(p-1)x2+q.x—p.

Ans. x2-1.

LEAST COMMON MULTIPLE.

ART. 109. A multiple of a quantity is any quantity that contains it exactly. Thus, 6 is a multiple of 2 or of 3; and ab is a multiple of a or of b; also, a(b—c) is a multiple of a or of (b-c).

ART. 110. A common multiple of two or more quantities, is a quantity that contains either of them exactly. Thus, 12 is a common multiple of 2 and 3; and 20xy is a common multiple of 2x and 5y.

ART. 111. The least common multiple of two or more quantities, is the least quantity that will contain them exactly. Thus, 6 is the least common multiple of 2 and 3; and 10xy is the least common multiple of 2x and 5y.

REMARK.-Two or more quantities can have but one least common multiple, while they may have an unlimited number of common multiples.

ART. 112. To find the least common multiple of two or more quantities.

From the nature of the least common multiple of two or more quantities, it is evident that it contains all the prime factors of each of the quantities once, and does not contain any prime factor besides; for, if it did not contain all the prime factors of any quantity, it would not be divisible by that quantity; and if it contained any prime factor not found in either of the quantities, it would not be the least common multiple. Thus, the least common multiple of ab and bc must contain the factors a, b, c, and no other factor. Hence,

The least common multiple of two or more quantities, contains all the prime factors of those quantities once, and does not contain any other factor.

With this principle let us find the least common multiple of mx, nx, and m2nz.

Arranging the quantities as in the margin, we see that m is a prime factor common to two of them. It must, therefore, even if found in only one of the quantities, be a factor of the least common multiple, and we place it on the left of the quantities. Then, since