CHAPTER VIII.

LEAST COMMON MULTIPLE.

154. A Multiple of a quantity is the product of the quantity and an entire factor.

155. A Common Multiple of two or more quantities is one into which each of them can be exactly divided.

156. The Least Common Multiple of two or more quantities is the least quantity into which each of the given quantities can be exactly divided.

L. C. M. stands for "Least Common Multiple."

157. I. When the quantities can be factored by inspection.

158. It is evident that the Multiple of a quantity contains all the factors of that quantity.

Also, that a Common Multiple of two or more quantities contains all the factors of the given quantities.

Also, that the Least Common Multiple of two or more quantities contains all the factors of the given quantities and no others.

159. Hence, if several quantities are resolved into their simplest factors, and each is taken the greatest number of times it occurs in any given expression, the product of these will be the Least Common Multiple.

(1) Find the L. C. M. of 25 ab and 35 a3b2. 25 a b = 5 x 5 xaxaxb;

35 a3b2 = 7 x 5 x a xa xa xbxo.

Here 5 occurs twice in one quantity, 7 once, a three times,

b twice; .. L. C. M. = 5 × 5 × 7 × a3 × b2 = 175 a3b2.

(2) Find L. C. M. of a2- b2, (a - b), and a3- b3. a2b2= (a - b) (a + b);

(a — b)2 = (a — b) (a − b) ;

a3 b3 (a - b) (a2 + ab + b2);

=

.. L. C. M. (a - b) (a - b) (a + b) (a2 + ab + b2)

=a5a2b3 — a3b2 + b3.

EXERCISE 44.

Find the L. C. M. of the following:

1. 14 a2+5a-1, and 6a2-a-2.

2. 12 a2 - 28 a 24, and 12 a 42 a24.

3. x2-3xy+2y, and x2-xy - 2 y2.

4. 6 x2y xÿ2-1, and 2xy + 3 xy2 — 2.

5. a2-5 am+6 m2, 2 a2 −7 am+3 m2, and 2 a2 −5 am +2 m2.

- 6, m3 + m2 - 17 m + 15, and m3

6. m3-6 m2 + 11 m

m2

7. 6x1+5x3y-6 x2y2, 4 x2+4x2y-3 xy2, and 4x2 — 6 xу — 18 y2.

8. a3 — b3, a3 + b3, and a1 + a2b2 + b1.

9. x2+ y2+2xy + 2 xz + 2 yz + z2, and x2 + y2 — z2 + 2 xy.

10. +4, x+4+x, and 8x+8a+ 32x2.

CHAPTER IX.

REDUCTION OF FRACTIONS.

160. A Fraction is an indicated division.

a

Thus, is a fraction, and indicates that a is divided by b.

161. The quantity above the line is called the numerator; the quantity below the line is called the denominator.

The numerator and denominator are the terms of the fraction.

162. As multiplying both dividend and divisor by the same quantity is equivalent to multiplying the fraction by unity, the value of the fraction is not changed thereby.

m

m

α

am

bm

Thus, multiplying both terms of by m, we have but am equals unity; ..

--

a
b bm

163. The reduction of a fraction is the process of changing its form without changing its value.

164. When the numerator and denominator of a fraction have no common factor, the fraction is in its lowest terms.

Hence, to reduce a fraction to its lowest terms, resolve both numerator and denominator into their simplest factors, and cancel all that are common to both.

Reduce the following to lowest terms:

34 a2b2c4 17 a2b2c x 2 c3

51 a+b2c

=

(1)

=

2c3 3a2

If the terms of the fraction cannot be readily factored by inspection, find the H. C. F. by the method of division.

Dividing (a — 2) into both terms of the fraction, we have

165. An improper fraction may be reduced to a whole or mixed number by performing the division indicated. If there is a remainder, this should be placed over the denominator and annexed to the integral part of the quotient.