107. To find the Greatest Common Measure of three quantities, a, b, c, find d the Greatest Common Measure of a and b; and the Greatest Common Measure of d and c is the Greatest Common Measure required.

Because every common measure of a, b, and c, measures d and c; and every measure of d and c measures a, b, and c (Art. 105); therefore, the Greatest Common Measure of d and c must be the Greatest Common Measure of a, b, and c.

108.

In the same manner the Greatest Common Measure of four or more quantities may be found.

The Greatest Common Measure of four quantities, a, b, c, d, may be found by finding a the Greatest Common Measure of a and b, and y the Greatest Common Measure of c and d; then the Greatest Common Measure of x and y will be the common measure required.

109. It should be borne in mind that the Greatest Common Measure of two or more algebraical quantities found as above is not necessarily the Arithmetical Greatest Common Measure of the same quantities when numerical values are given to the letters contained in them; and the reason is this there may be factors in the several quantities which, in their algebraical state, are prime to one another, but become divisible by some common number, when certain values are given to the letters contained in them. For example, the factors x-7, and a-4, have no common measure greater than 1, as algebraical expressions; but if 10 be put for x, they become 3, and 6, which have a common measure, 3. Hence it is plain, that if we exclude certain factors from the Greatest Common Measure as having no common divisor, and afterwards change their form so as to make them to have a common divisor, the Greatest Common Measure obtained on the former supposition cannot possibly agree with the Greatest Common Measure obtained on the latter supposition. In fact, the phrases 'Greatest Common Measure,' 'Lowest Terms, Least Common Multiple,' applied to algebraical quantities, do not regard comparative magnitude.

110. In practice the Greatest Common Measure of two or more algebraical quantities is frequently found by a more expeditious method than the preceding, as follows. Taking Ex. 2, Art. 105.

Ex. Required the G. c. M. of a‘-x1 and a3-a2x—ax3+x3.
First, a-(a2+ x2)(a3— x2).

Also a3-a x-ax2+x3 = a2(a−x)-x' (a-x),

= (a2— x3)(a−x) ;

therefore a2x2 is a common factor or divisor of the proposed quantities; and since the other factors a2+ and a-r have no common measure greater than 1, therefore a2-x2 is the Greatest Common Measure required.

111. In the same manner fractions are usually reduced to their lowest terms without the application of the Rule for finding the G. C. M. of the numerator and denominator.

=

(a+b)(a+b+c)(a+b−c)

4b3c2— (a2—b3—c2)2 ̄ ̄ (2bc+a*—b2—c3) (2bc−a2+l3+c2) '

112. Fractions may be changed to others of equal value, with a common denominator, by multiplying each numerator by every denominator except its own, for the new numerator, and all the denominators together for the common denominator.

= ;

adf cbf ebd
bdf' bdf' bdf'

= ; and

are

ebd e

=

fractions of the same value respectively with the former, having the adf a cbf c common denominator bdf. For bdf b bdf d' bdf f (Art. 101); the numerator and denominator of each fraction having been multiplied by the same quantity, viz. the product of the denominators of all the other fractions.

113. When the denominators of the proposed fractions are not prime to each other, find their Greatest Common Measure; multiply both the numerator and denominator of each fraction by the denominators of all the rest, divided respectively by their Greatest Common Measure; and the fractions will be reduced to a common denominator in lower terms than they would have been by proceeding according to the former rule.

114. To obtain them in the lowest terms each must be reduced to another of equal value, with the denominator which is the Least Common Multiple, or Lowest Common Multiple, of all the denomi

nators.

It becomes necessary therefore to investigate a rule for finding the Least Common Multiple of two or more quantities. And first,

115. To find the Least Common Multiple of any number of simple quantities.

To do this, observe the combinations of letters which form the several quantities, and resolve each quantity, by inspection, into its simple factors. The object then will be, to construct such a quantity as shall contain all the different factors found in the proposed quantities, but no factor repeated

which is not also similarly repeated in some one of them; for thus we shall obviously form a quantity, and the least quantity, which is divisible by each of the proposed quantities without remainder, that is, the Least Common Multiple of them all. To this end, detach from each quantity all the factors, which are common to two or more of them, until the quantities are left prime to each other. The continued product of these common factors and prime results will be the Least Common Multiple required.

Thus, let the L.C.M. of 2a, 6ab, and 8ab be required. We see that the three quantities have a common factor 2a, which being detached leaves the quantities 1, 3b, and 46: of these again, the two latter have a common factor b, which being detached leaves the quantities 1, 3, and 4; and these are prime to each other. Therefore the L.C.M. required is

2axb×1×3×4, or 24ab.

OBS. Since the detaching of the Common Factors is the same thing as dividing the quantities by their Greatest Common Measures, it is clear that this method coincides with the arithmetical rule given in Art. 23.

It may also be observed that the preceding method is applicable to compound quantities, as well as simple, provided that each of the quantities can be readily resolved into its component factors. Thus, if the L.C.M. of abad, and ab-ad be required, we see that the quantities have a common factor a, and when stripped of this become b+d, and b- d, which are prime to each other. Therefore the L.C.M. required is

a(b + d)(b − d) or ab2 — ad2.

The following method is generally applicable to all quantities Simple or Compound.

116. To find the Least Common Multiple of two quantities, or the least quantity which is divisible by each of them without remainder.

Let a and b be the two quantities, their greatest common measure, m their least common multiple, and let m contain a, p times, and b, q times, that is, let m = pa = qb; then dividing the

two latter equal quantities by pb (Art. 82),

a

q

=

b

p

; and since m

is the least possible, p and q are the least possible; therefore

α

α

-; hence

is the fraction in its lowest terms*, and consequently q=~-~

b

=

x

* For, if not, let some other fraction be the fraction in its lowest terms; then since

α

" b

Ρ

multiplying these equal quantities by p'b (Art. 81), q'b=p'a, or there are common multiples of a and b less than pa and qb, which is impossible, since pa and qb are the least. ED.

The rule here proved may be thus enunciated:

Find the G.C.M. of the two proposed quantities; divide one of them by this G.C.M.; and multiply the quotient thus obtained by the other quantity. The product is the Least Common Multiple required.

Ex. Required the Least Common Multiple of a1-x' and a3-a2x

The G.C.M. of these two quantities (See Art. 105, Ex. 2), is a2-x2; and (a*-x1)÷ (a3— x3) = a2+x3. Therefore the Least Common Multiple required

=(a'+x3).(a3— a3x— ax3+x3),

=a3—a*x—ax*+x3.

117. Every other common multiple of a and b is a multiple of m.

Let n be any other common multiple of the two quantities; and, if possible, let m be contained r times in n, with a remainder 8, which is less than m; then n-rm=8; and since a and b measure n and rm, they measure n-rm, or 8 (Art. 104); that is, they have a common multiple less than m, which is contrary to the supposition.

118. To find the Least Common Multiple of three quantities a, b, c, find m the Least Common Multiple of a and b, and n the Least Common Multiple of m and c; then n is the Least Common Multiple sought.

For every common multiple of a and b is a multiple of m (Art. 117); therefore every common multiple of a, b, and c is a multiple of m and c; also every multiple of m and c is a multiple of a, b, and c; consequently the Least Common Multiple of m and c is the Least Common Multiple of a, b, and c.

And similarly if there be four or more quantities of which the Least Common Multiple is required.

Ex. Required the Least Com. Mult. of x3- a3x—ax2+a3, x1—a1, and ax3+ a3x-a2x2— a1.

Here ar3+a3x-a2x2— a1= a(x2+ a3x — ax3—a3);

.. to find the G.c.M. of this quantity and the first, reject the factor a;