SECTION X. GREATEST COMMON DIVISOR.* 72. A COMMON DIVISOR of two or more quantities is any quantity that will divide each of them without remainder. 73. THE GREATEST COMMON DIVISOR of two or more quantities is the greatest quantity that will divide each of them without remainder. 74. To deduce a rule for finding the greatest common divisor of two or more quantities, we demonstrate the two following theorems: THEOREM I. A common divisor of two quantities is also common divisor of the sum or the difference of any multiples of each. α Let A and B be two quantities, and let d be their common divisor; d is also a common divisor of m A ±n B. That is, d is contained in m A + n B, m p + n 9 times, and in n B, mp-n q times; i. e. d is a common divisor of the m A THEOREM II. The greatest common divisor of two quantities is also the greatest common divisor of the less and the remainder after dividing the greater by the less. Let A and B be two quantities, and A > B; and let the process of dividing be as appears in the margin. Then, as the dividend is equal to the product of the divisor by the quotient plus the remainder, B) A (q r And, as the remainder is equal to the dividend minus the product of the divisor by the quotient, Therefore, according to the preceding theorem, from (1) any divisor of r and B must be a divisor of A; and from (2) any divisor of A and B, a divisor of r; i. e. the divisors of A and B and B and r are identical, and therefore the greatest common divisor of A and B must also be the greatest common divisor of B and r. In the same way the greatest common divisor of B and r is the greatest common divisor of r and the remainder after dividing B by r. Hence, to find the greatest common divisor of any two quantities, RULE. Divide the greater by the less, and the less by the remainder, and so continue till the remainder is zero; the last divisor is the divisor sought. NOTE 1. The division by each divisor should be continued until the remainder will contain it no longer. NOTE 2.—If the greatest common divisor of more than two quantities is required, find the greatest common divisor of two of them, then of this divisor and a third, and so on; the last divisor will be the divisor sought. NOTE 3. The common divisor of x y and xz is x; x is also the common divisor of x and xz, or of a x y and xz; i. e. the common divisor of two quantities is not changed by rejecting or introducing into either any factor which contains no factor of the other. NOTE 4. It is evident that the greatest common divisor of two quantities contains all the factors common to the quantities. CASE I. 75. To find the greatest common divisor of monomials. 1. Find the greatest common divisor of 8 a2 b33 c d, 16 a3 b3 c2, and 28 aa ba c. The greatest common divisor of the coefficients found by the general rule is 4; it is evident that no higher power of a than a3, of b than b3, of c than itself, will divide the quantities; and that d will not divide them; therefore, the divisor sought is 4 a2 b3 c. Hence, RULE. Annex to the greatest common divisor of the coefficients those letters which are common to all the quantities, giving to each letter the least exponent it has in any of the quantities. 2. Find the greatest common divisor of 63 a3 b1 ca d®, 27 at 68 c5, and 45 a2 b9 c3 d. Ans. 9 a2 b7 c3. 3. Find the greatest common divisor of 75 x1 y5 z3 and 125 a b x3 y1 z3. 4. Find the greatest common divisor of 99 a b2 c2 d3 xo y3 and 22 a2b4 c3 do x5. Ans. 11 a b2 c3 d3 x5. 5. Find the greatest common divisor of 17 x1 y2, 19 x2 y3, and 212 bx y1 z5. CASE II. 76. To find the greatest common divisor of polynomials. 1. Find the greatest common divisor of x2 — y2 and x2-2xy + y2. Arrange the terms of both quantities in the order of the powers of some letter, and then proceed according to the general rule in Art. 74. NOTE 1. - If the leading term of the dividend is not divisible by the leading term of the divisor, it can be made so by introducing in the dividend a factor which contains no factor of the divisor; or either quantity may be simplified by rejecting any factor which contains no factor of the other. (Art. 74, Note 3.) a, NOTE 2. Since any quantity which will divide a will divide and vice versa, and any quantity divisible by a is divisible by a, and vice versa, therefore all the signs of either divisor or dividend, or of both, may be changed from + to —, or to +, without changing the common divisor. NOTE 3. When one of the quantities is a monomial, and the other a polynomial, either of the given rules can be applied, although generally the greatest common divisor will be at once apparent. 2. Find the greatest common divisor of ax 2 a c x3 +4 a2 c x2 — 6 a3 c x - a2x6 8a5x8 · 8 a3 x3 2 c x2 — 2 a c x23 + 4 a2 c x2 — 6 a3 c x — 20 aa c x1 — a x23 — 8 a1 3. Find the greatest common divisor of a1 a3 a2 xa x2- x3. + x1 and Ans. a2 x2. 4. Find the greatest common divisor of a* a5 a3 x2. x4 and Ans. a2 - x2. 5. Find the greatest common divisor of 2 a x2 — a2 x and 2x2+3 a x + a2. Ans. 3 ax 4 a. 6. Find the greatest common divisor of 6 a x and 6 a x3 a x2 12 ах. 7. Find the greatest common divisor of x* 23+ y3. 8. Find the greatest common divisor of 3 x3 and 2 x3 16 x - 6. 9. Find the greatest common divisor of x3 x2- y2. 10. Find the greatest common divisor of 10 x3 +30 y3 and x3 + 2 x2 y + 2 x y2 + y3. Ans. xy. 11. Find the greatest common divisor of a + a3 + a2 12. Find the greatest common divisor of 7 a x2 + 21 a x3 +14a and x + x2 + 2x3 x. Ans. x+1. - 8 a6 y 13. Find the greatest common divisor of 27 a3 ya . and 3y-2 ay + 3 a2 y — 2 a3 y1. Ans. 3y22ay. 14. Find the greatest common divisor of a3 + a 10 and a1 16. Ans. a 2. NOTE 5. The greatest common divisor of polynomials can also be found by factoring the polynomials, and finding the product of the factors common to the polynomials, taking each factor the least number of times it occurs in any of the quantities. (Art. 74, Note 4.) 15. Find the greatest common divisor of 3 a x2. 3axy 4 ay and a3 x-x+a3 y — y. 3 4ax + 3 a x2-4 ax+3axy-4a y = a(x + y) (3x-4) a3 x x + a3 y y = (x + y) (a− 1) (a2 + a + 1) Ans. xy. |