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have also a common measure, that same number or quantity will measure a, b, and c; and if x be the greatest common measure of d and c, it will also be the greatest common of a, b and c.

And, in like manner, if there be any number of quantities; a, b, c, d, &c.; and that x is the greatest common measure of a, and b; y the greatest common measure of x and c; z the greatest common ineasure of

y

and d; &c. &c.; then will y be the greatest common measure of a, b, and c; : the greatest common measure of a, b, c, and d; &c. &c.

134. The preceding method of demonstration is similar to that given by Bridge in his Treatise on the Elements of Algebra : The following is according to the manner of GARNIER. Thus, to find the greatest common divisor of any number of quantities A, B, C, &c., it is sufficient to know the method of finding the greatest common divisor of two numbers 'or quantities. For this purpose, we will at first seek the greatest common divisor D of the quantities A and B, then the greatest common divisor D of D' and C,

and finally the last greatest common divisor will be that which was required.

Let, in order to demonstrate it, the three quanti ties be A, B, C; we will have SA=mD,

A=mrD,

;
1 st
B=nD,

B=nrD',

whence
Š D=rD',
2d
C=qD,

C=qD;J

'. m and n are necessarily prime to one another,otherwise D would not be the greatest common divisor of A and B; rand q are also prime to one another, in order that D' may be the greatest common divisor of D and C. Now rD', the greatest common divisor of A and B, cannot be the greatest common

and so on,

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divisor of A, B, and C, unless that r be equal to 4,

9 or a factor of q; but r and q being prime to one another; D', remains the greatest common divisor of A, B, and C.

135. As the problem of finding the greatest common divisor of any two quantities A and B, is the

A same as to reduce a fraction

to its most simple

B expression; because that in dividing A and B by their greatest common divisor, we have the two least quotients possible ; admitting this enunciation, and supposing >B.

The greatest common divisor of A and B, cannot exceed B; it could be B itself, which we can readily know, if we perform the division of A by B, which gives А R

• (1), 9

B B q being the integral quotient, and R the remainder, if A is not exactly divisible by B. The fraction A

R being changed into 9+

q cannot be reduced unB

B'
R

B less that

or its reciprocal is reducible, beB

R cause q is an integral quantity which is always irreducible; or B being >R, the quantity which ought

B B to reduce cannot exceed R, it might be R itself,

R' which we will know in performing the division of B by R, which gives

B

R
R

(2),
a't

R 9 being the integral part of the quotient, and R' the remainder <R; we say still that the reduction of

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9

=

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R"=9"' +

or

or

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or

B

R depends on that of or its reciprocal, because R

R' that q' is an irreducible quantity; so that by continuing in this manner we shall have the following decompositions :

R R"

=9"+ (3),
R R'
R
R"

(4).

R" We see very clearly that the quantity which ought А

R B to reduce is that which must reduce B

B R R R which must reduce

which must reduce

R R" R"

R' R

R” If, for example, R"=0, this quantity cannot be greater than R": R" is therefore the greatest quan

A tity which can reduce the fraction Bi consequently

В it is the greatest common divisor of A and B.

136. Let R"=0 and R=1: unity will be, according to what has been above demonstrated, the greatest common divisor of A and B ; the fraction А

will therefore itself be the greatest expression, B that is, it will be irreducible. Reciprocally, the last divisor being unity, we may conclude that the fruction proposed is irreducible, or in its lowest terms.

137. It may also be shown, that the greatest common measure of two quantities will, in no respect, be altered, by multiplying or dividing either of them by any quantity which is not a divisor of the other, or that contains no factor which is common to both of them; thus, let the quantities ab and ac be taken, of which the common measure is a; theri. if ab be multiplied by d, they will become abd, and ac; where it is evident that a is the common measure, as before. And, conversely, if the first of the two quantities abd, ac, be divided by d, they will · become ab, ac, where a is still the common measure.

138. But it will not be the same if one of the two quantities be multiplied or divided by a quantity which is a divisor of the other, or has a common factor with it; for if the first of the two quantities ab, ae, be multiplied by c, they will become abc, ac, of which the common divisor is ac, instead of a; and, conversely, if the first of the two quantities abc and ac, be divided by c, they will become ab and ac; of which the common divisor is a, instead of ac.

139. Hence, if the numbers or quantities be mncN, pqcN'; the common factor c, to simplify the operation, may be suppressed, observing, in the meantime, after having found the greatest common divisor a, of the two quotients N and N', to multiply it by this factor c, and the product will be the greatest common divisor sought. Also, if a factor d is introduced into the two quantities, it is necessary to divide the greatest common divisor by this factor.

140. As the foregoing demonstration may be extended to any algebraic quantities whatever, we are therefore conducted to this practical rule. To find the greatest common divisor of two or more

compound algebraic quantities.

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RULE.

141. Arrange the two quantities according to the order of their powers, and divide that which is of the highest dimensions by the other, having first expunged any factor that may be continued in all the terms of the divisor without being common to those of the dividend; then divide this divisor by the re

mainder, simplified, if necessary, as before; and so on, for each remainder and its preceding divisor, till nothing remains : then the divisor last used will be the greatest common divisor required. And the greatest common divisor, of more than two compound quantities, is found in like manner; by finding in the first place the greatest common divisor of two of them, as above, and then of that common divisor and the third, and so on. The last divisor, thus found, will be the greatest common divisor of all the quantities.

EXAMPLE 1. The greatest common divisor of the compound quantities 3a3- 3a2b+aba -63 and 4a2b - 5ab2 +-63, is required. Dividend.

Divisor. 3a3 ... 3a2b+ ab(4a2b--5ab2 +5%)----= За 4

4a - 5ab +62 1223 -12a2b+-4ab2 - 463 12a - 15a2b+3aba

Partial quot. 30

3

(3a2 b +-a/2 -463)=-=
3a tab- 462
4

Divisor.
12a2 + 4ab-1662 4a2 --5ab +62
12a2 - 15ab-t 362

Partial quot. 3
19ab-1962
Dividend.

Divisor,
4a? - 5ab +62 (19ab-1962-196=
4a? - tab

a-6
ab +62
th+63 | Quot. 4a-

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