THE LEAST COMMON MULTIPLE OF TWO OR MORE QUANTITIES. 53. A multiple of a quantity is one which can be divided by it without remainder; thus 8ax is a multiple of 2a, and 16 x y3 is a multiple of 4 x y2, or of 2 x y3.

54. A common multiple of two or more quantities is one which can be divided by each of them without remainder. Thus 36 ax is a common multiple of 2 a and 9x.

55. The least common multiple of two or more quantities is the least quantity which can be divided by each of them without remainder. Thus the least common multiple of 8 a3 and 12 a* x3 is 24 a* x3; and the least common multiple of 6a2x2y3, 15 a3 x y3, and 21 ax3y, is 210 a3 x3 y3.

56. Let a and b be two quantities, m their greatest common measure, and their least common multiple; then we have,

a=hm and b = km,

where h and k have no common measure, since m is the greatest common measure of a and b; hence hk is the least common multiple of h and k; therefore the least common multiple of hm and km is ħ km; consequently the least common multiple of a and b is

Hence the least common multiple of two quantities a and b is found by dividing their product ab by their greatest common measure; or, which amounts to the same thing, divide either of the quantities by their greatest common measure, and multiply the quotient by the other quantity.

This is evident, since

ахь a

m

b

= xb== ха.

57. Let a, b, c be three quantities, and the least common multiple of a and b; then the least common multiple of 7 and c is the least common multiple of a, b, and c. For any common multiple of a and b contains l, their least common multiple, and therefore every multiple of I is a common multiple of a and b, and every common multiple of aud e is a common multiple of a, b, and e; consequently the least common multiple of and c is the least common multiple of a, b, and c.

To find the least common multiple of three quantities: find the least common multiple of two of them, and then the least common multiple of this last multiple and the third quantity will be the least common multiple of all three, and so on, if there are four or more quantities.

EXAMPLES.

1. Find the least common multiple of 15 x2 y2, 6 x3y, and 12 x y3. The greatest common measure of 15 x2 y2 and 6 x3y is (47) 3x2 y ; 15xy × 6x3 y = 5y × 6 x3 y = 30 x3 y2 the least common mul3xy tiple of 15 x2 y2 and 6x3y. Again, the greatest common measure of 30xy and 12 x y3 is 6xy; hence the least common multiple of all 30 x3 y3

three is

6 x y3

× 12xy = 5 x x 12 x y = 60 x3 y3.

.. 3xy × 2 × × × y × 5 × × × 2y = 60 x3 y3, as before.

This last method is similar to that employed in arithmetic for finding the least common multiple of any number of quantities.

2. Find the least common multiple of 8x (xy), 15 x (x − y)2 and 12 x3 (x2- y2).

.. x2 (x − y) × 3

= 120 x (x2 - y)

× 4 × 2 × 5 x2 (x − y) × (x + y)

(xy), the least common multiple.

3. Find the least common multiple of 3 x3-2x2 -x and 6x2-x-1. Suppressing the simple factor x in the former quantity, we have

Hence 3x+1 is the greatest common measure, and therefore

x x (6 x2-x-1) = x (x − 1) (6 x2 — x − 1)

= 6 x1 — 7 x3 + x = least common multiple.

=

Find the least common multiple

4. Of 8x, 10 x3y, and 12 x2 y3.

5. Of 10 (x2 + xy), 8 (xy — y2), and 5 (x2 — y2).

Ans. 120 x*y*.

Ans. 40 x y (x − y3).

6. Of 2 a2 (a + x), 4 a x (a − x), and 6 x2 (a2 − x2).

Ans. 12 ax (a2 — x2).

and a3 + a x3

x y* + y3.

a x2 + x*.

a2x2 + (a2 + a) x3 − x3.

FRACTIONS.

58. The principles of algebraic fractions are precisely similar to those of arithmetical ones, and the management of fractions consists either in certain transformations of them to others of the same value, or in the usual processes of addition, subtraction, multiplication, and division.

59. If denote an algebraic fraction, then b denotes the number of equal parts into which the unit is supposed to be divided, and a the number of such parts to be taken; or we may conceive the quantity represented by a to be divided into b equal parts, and one of them to be

Hence, if the numerator of a fraction be increased any number of times, the fraction itself is increased as many times; therefore

Again, if the denominator be diminished any number of times, the magnitude of the equal divisions of the unit will be increased as many times, and consequently the same number of the increased divisions being taken, the fraction will still be increased as many times; hence

We have therefore this principle: a fraction is multiplied by a quantity, either when its numerator is multiplied, or its denominator is divided by that quantity.

Again, if the numerator of a fraction be diminished any number of times, the fraction itself is diminished as many times; therefore

Or, if the denominator of a fraction be increased any number of times, the number of divisions of the unit will be increased as many times, and the magnitude of the equal divisions will be diminished as many times, and consequently the same number of the diminished divisions being taken, the fraction will still be diminished as many times; hence

We have therefore another principle: a fraction is divided by a quantity, either when its numerator is divided or its denominator is multiplied by that quantity.

Lastly, if the numerator of a fraction be increased any number of times, the fraction will be increased as many times, and if the denominator be increased as many times, the fraction will be diminished the same number of times as it was before increased; hence its value will not be altered. In a similar manner, it may be shown that if the terms of a fraction are both divided by the same quantity, its value is not altered.

We have then a third principle: a fraction is neither multiplied nor divided by a quantity, if both its terms are either multiplied or divided by that quantity; that is, the fraction remains unaltered in value, though the forms of its terms are changed by equal multiplication or division.

Any algebraical symbol a may be represented in a fractional form by writing it as the numerator of a fraction whose denominator is 1; thus 3x2 (a + x) @= 2x= and 3x (a + x) = 1

a

2 x

REDUCTION OR TRANSFORMATION OF FRACTIONS.

60. To reduce a fraction to its lowest terms, or its simplest form.

The value of a fraction is not altered, if both its terms are divided by the same quantity (59); hence it is evident that to reduce any fraction to its lowest terms, is to divide its terms by their greatest common measure. If the terms of the fraction have no common measure other than unity, the fraction is already in its simplest form. Thus

ma

mb

α

= ㄜˊ by dividing both terms by m, their greatest common measure.

to its lowest terms.

Here 12x2 - 15 x y + 3y = 12 x 12 x y (3 x y -3y2) = 12x (xy) 3y (xy) = (x − y) x (12x-3y); and 6x-6xy + 2xy - 2y = 6 x2 (x − y) + 2y2 (x − y) = (xy) x (6x+2y);

therefore

12x2 - 15 x y +3y2

=

(xy) x (12x-3y)

(xy) × (6x2 + 2 y3)

3 (4x - y) = 2(3x2 + y2)`'

In this example we have found the greatest common measure, x of the terms of the fraction, without having recourse to the usual method, which is seldom required in practice, and is frequently a very tedious operation.

Reduce each of the following fractions to its lowest terms:

x + y

m + 2 n'

61. To reduce a fraction to the form of a mixed quantity, when the

reduction can be effected.

n

If the numerator of the fraction

can be divided by the denominator

с

c, and leave a remainder b, then the form of the numerator will be ac+b; hence

ac + b

с

ac- b

b

с

b

с

=3

= a +

1120

Also

с

C

16

15

Thus

=

5

5

9 a

a2

2a+

4

5'

or

9 a2

b

с

16

5

=

=3a2

a mixed quantity.

This transformation or reduction is nothing else than dividing the numerator by the denominator, as in common division.

to a mixed quantity; that is, divide a2 +x2 by