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RULE.

Divide one of the quantities by their greatest common divisor, and multiply this quotient by the other quantity, and the product will be their least common multiple.

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NOTE 1. If the least common multiple of more than two quantities is required, find the least common multiple of two of them, then of this common multiple and a third, and so on; the last common multiple will be the multiple sought.

NOTE 2. In case the least common multiple of several monomials and polynomials is required, it may be better to find the least common multiple of the monomials by the Rule in Case I., and of the polynomials by the Rule in Case II., and then the least common multiple of these two multiples by the latter Rule.

1. Find the least common multiple of x2 x2-2xy + y2.

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y2 and

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y, with which we divide one of

the quantities; and multiplying the other quantity by this quotient, we

have the least common multiple (x2 — y2) (x − y).

2. Find the least common multiple of 2 a2x2, 4 x2 y, x1, and a5 · a3 x2.

a1

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The least common multiple of the monomials is 4 ay; and the least common multiple of the polynomials is a3 (aa — x1).

The greatest common divisor of these two multiples is a2; and dividing one of these multiples by a2, and multiplying the quotient by the other, we have 4 a3 x2y (a* - - x) as the least common multiple.

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3. Find the least common multiple of 3 a2 b3, 6 a2 by, a8, and a2-4a4.

Ans. 6 a2 by (a3 — 8) (a — 2).

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4. Find the least common multiple of 3 x3 24 x

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5. Find the least common multiple of a1 — x1 and a — x3.

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6. Find the least common multiple of x 1, x2+2x+1, and (1)2. Ans. 6x — x2 + 1.

7. Find the least common multiple of xa — y and x3 + y3. 8. Find the least common multiple of a3 + a 10 and

a1- 16.

NOTE 3. - The least common multiple of any quantities can also be found by factoring the quantities, and finding the product of all the factors of the quantities, taking each factor the greatest number of times it occurs in any of the quantities. (Art 80.)

9. Find the least common multiple of 2—2xy + y2, xy, and (x + y)2.

x2 - 2xy + y2 = (x − y) (x − y)

(x2 + y2) (x + y) (x − y)

x1 — y1 =

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Hence L. C. M= (x − y) (x − y) (x2 + y2) (x+y) (x+y)

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10 Find the least common multiple of 3 a x2 3axy-4ay and a3 x-x+a3y—y.

(See 15th Example, Art. 76.)

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Ans. a(x+y) (3x-4) (a2 + a + 1) (a−1) = 3a1x2 —

4a+x+3axy-4aty

3ax2+4ax-3axy + 4ay.
3axy+4ay.

SECTION XII.

FRACTIONS.

82. WHEN division is expressed by writing the dividend over the divisor with a line between, the expression is called a FRACTION. As a fraction, the dividend is called the numerator, and the divisor the denominator.

Hence, the value of a fraction is the quotient arising from dividing the numerator by the denominator.

xy y

Thus, is a fraction whose numerator is xy and denominator y, and whose value is x.

83. The principles upon which the operations in fractions are carried on are included in the following

THEOREM.

Any multiplication or division of the numerator causes a like change in the value of the fraction, and any multiplication or division of the denominator causes an opposite change in the value of the fraction.

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which is y times the value of the given fraction.

Dividing the numerator by y,

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2d. Changing the denominator.

Multiplying the denominator by y,

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COROLLARY.

which is y times the value of the given fraction.

·Multiplying or dividing both numerator and denominator by the same quantity does not change the value of the fraction.

For if any quantity is both multiplied and divided by the same quantity its value is not changed.

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84. Every fraction has three signs: one for the numerator, one for the denominator, and one for the fraction as a whole.

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If an even number of these signs is changed from to —, or to +, the value of the fraction is not changed; but if an odd number is changed, the value of the fraction is changed from to, or to +.

+

Thus, changing an even number,

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The various operations in fractions are presented under the following cases.

CASE I.

85. To reduce a fraction to its lowest terms.

NOTE. A fraction is in its lowest terms when its terms are mutually prime.

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divide both terms by any factor common to them, as 4 a2; and both terms of the resulting fraction by any factor common to them, as 2xy; or we can divide both terms of the given fraction by their

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Divide both terms of the fraction by any factor common to them; then divide these quotients by any factor common to them; and so proceed till the terms are mutually prime. Or,

Divide both terms by their greatest common divisor.

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