What is the Least Common Multiple of 2ab and 362? Of 4x and 4y2? Of 5a and 3a2b? Of ab and 2bc2? Of 7ax and ax2? Of 3a2 and 6ay2? Of axy and 5x2y2?

(72.) The least common multiple of two or more quantities is composed of the smallest selection of factors that includes the factors of each given quantity.

For example, take the quantities 3ab2c and 6a2bxy,

Resolving these two quantities into their prime factors, we have

3abbc and 3×2 aab xy.

If we take 3×2 aa bbc xy, we shall have the smallest selection of factors that includes the factors of each of the two given quantities.

Then the product 6a2b2cxy is the least common multiple, because it is the smallest quantity that each of the given quantities will divide, without a remainder.

(73.) The least common multiple of two quantities, is equal to their product divided by their greatest common measure.

For since the greatest common measure of two quantities, is composed of all the factors which are common to those quantities, (64), these factors will enter twice in the product of the quantities.

If, therefore, the product be divided by the greatest common measure, the quotient will contain only those factors which are common to the two quantities, and those which are peculiar to each of them; and these are the factors of the least common multiple, (72).

RULE VIII.

(74.) To find the Least Common Multiple of Two or more Quantities.

1. Set the quantities in a line, from left to right, and divide any two or more of them by any prime quantity, greater than unity, that will divide them, without a remainder,-placing the quotients and the undivided quantities in a line below.

2. Divide any two or more of the quantities in the lower line, as before; and so on, until no two quantities in the lowest line can be so divided. The product of the divisors and quantities in the lowest line, will be the least common multiple required.

3. If no two of the given quantities can be divided as above, the product of all the quantities will be their least common multiple.

In the first operation we divide 3ax2 and 6a2y by 3, and set down 3y-4y3 without dividing it; and in like manner in the second operation. In the third operation we set down x2 without dividing it.

Then 3ayx2×2a × (3—4y2), equal to 18a2x2y-24a2x2y3, is the least common multiple of the three given quantities.

This Rule depends on proposition (72): the divisors and quantities in the lowest line, are the smallest selection of factors that includes the factors of each given quantity.

EXERCISES.

1. Find the least common multiple of

ax2, 2a2y, 4y+y2, and ax2+4x2,

Ans. 8a3x2y + 2a3x2y2+32a2x2y+8a2x2y2.

4. Find the least common multiple of
4y3, 2uy2, 5x-5ab, and 10a-5.

Ans. 40a2y3-40a2by3-20ay3+20aby3.

5. Find the least common multiple of

5a, 10ab, 3y+3y2, and 6y3+3y2.

Ans. 90aby3+60aby*+30aby2.

6. Find the least common multiple of

4, 4x2, 8y-8, and 2x3-ax3.

Ans. 16x3y-16x3-8ax3y+8ax3

7. Find the least common multiple of 10, 5ax, 4a-8y2, and 2x+6x2. Ans. 20a2x-40axy2+60a2x2-120ax2y2

8. Find the least common multiple of

3y2, 12y, 5x2-10, and 4y-8y3.
Ans. 60x2y2-120y2-120x2y+240y*

9 Find the least common multiple of
14, 7a, 2y2-4, and 28+7ay.

Ans. 56ay2+112a+14a2y3-28a2y.

10. Find the least common multiple of
a1-x4, and a3—a2x-ax2+x3, (73).

11. Find the least common multiple of

Ans. a3 — a1x—ax1+x3.

x2+2bx+b2 and x3-b2x. (73).

Ans. x+bx3-b2x2-b3x.

12. Find the least common multiple of
a2-3ab+262 and a2-ab-262. (73).

Ans. a3-2a2b-ab2+263.

13. Find the least common multiple of
4a3-2a2-3a+1 and 3a2 —2a-1. (73).

Ans. 12a-2a3-11a2+1.

41

CHAPTER IV.

FRACTIONS.

(75.) An algebraic FRACTION represents the Quotient of its numerator divided by its denominator.

a

Thus represents the Quotient of a divided by 3x.

3x

In reading an algebraic Fraction, it will often be necessary to use the terms numerator and denominator, to avoid ambiguity in reference to the division which is expressed.

Thus if the Fraction

a+x
y

be read, a+x divided by y, it might be understood that only x is to be divided by y. But the true sense would be conveyed by saying, rumerator a+x, denominator y.

A Fraction is thus employed to represent the Quotient, when the divisor is not a factor of the dividend. The quotient in this case may also be represented by means of

Negative Exponents.

(76.) Any quantity with a negative exponent is equivalent to a unit divided by the same quantity with the sign of its exponent changed.

Thus a 2, a with exponent -2, is equivalent to

For a2a-2 is equal to ao, (41);

and by dividing each of these equals by a2,

1

What is the fractional equivalent of a-3? and how is it proved? Of x-4? and how is it proved? Of y-5? and how is it proved? Of (ax)-2? Of (x-y)-3 ?

(77.) When the Divisor is not a factor of the dividend, the Quotient may be represented by a fraction (75), or by the dividend multiplied into the divisor with the sign of its exponent changed.

a-x2 will give the quotient ax-2, because this quotient multiplied into the divisor x2, produces ax°, (41), which is equal to a, the dividend, (48).

We have, therefore, ax-2 for an integral, and

form of the quotient of a÷x2.

for a fractional

What is the integral form of the Quotient of a÷63 ? and how is it proved? Of xy? and how is it proved? Of 1÷a2 ? and how is it proved? Of a2÷x ? Of b-ac2 ? Of a÷5?

Transfer of Factors.

(78.) Any factor may be transferred from the denominator to the numerator, and vice versa, by changing the sign of its exponent. For example, if we divide a by x2y, fractionally,

If we divide a by the factors r2 and y, separately, we shall find

ax2 equal to ax-2, (77), and ax-2÷y equal to

ax-2

У

These two quotients being necessarily equal to each other, we see that a2 may be transferred from the denominator to the numerator, by changing the sign of its exponent.

If we also transfer the factor y, we shall have

If we transfer the factor a from the numerator a, or la, to the de nominator, we find

Under what different forms may the Quotient of a÷bx2 be
Of 3a-x3? Of 2c÷3a4? Of ab÷2y2 ?

represented?