Elements of algebra

was one-fifth of a mile less than he had supposed, and that to have won his wager he must have walked forwards two miles an hour faster than he did. What is his rate per hour back

wards? (1823).

12. A lent B a sum of money, to be repaid with interest at the end of a year, and received as security Spanish five per cent bonds to such an amount that their interest was equal to the interest of the debt. At the year's end B proved insolvent, and Spanish bonds having fallen 40 per cent, A found that he had lost £400. Had they not fallen in value he would have been enabled to repay himself, and to return to B £250; and had he been at liberty to have sold them out when they were at 50, which was before the interest on them was payable, he would have lost only £300. Required the amount of the debt, and its interest, and the price of Spanish bonds at the beginning and end of the year.

(1823).

13. A merchant wishing to buy a certain quantity of pimento, the price of which he calculates at the rate of five bags for £8, transmits to his foreign agent the requisite sum of money. Before the order arrives pimento has risen in value; and the money is sufficient only to buy a quantity less by 18 bags than that which the merchant intended. It appears also that as many bags as of the intended quantity, increased by 54, will now cost £10:7s. more than they would have done had the price not varied. What is the quantity purchased? (1824).

CHAPTER VII.

GREATEST COMMON MEASURE-LEAST COMMON MULTIPLEPROPERTIES OF NUMBERS.

121. In this chapter we propose to examine a few of the most simple properties of numbers. These properties are of two kinds, the one relating to abstract numbers, independently of the mode of their representation, the other depending on the notation which is adopted to represent them. We shall, in this chapter, treat of the former class.

PROP. If a whole number and a proper fraction be equal to a whole number and a proper fraction, then the whole numbers are equal, and the fractions are equal.

For, if ab, then by subtraction, we obtain a b = rp
P

i. e., a number not less than 1 is equal to a fraction less than 1, which is absurd.

122. PROP. To find the greatest common measure of two numbers, a and b; that is, the greatest factor which is common to them both.

Let a be divided by b with a remainder c,

d divides c (by equation 3),

qe and qc+d, or b (by equation 5):

it divides pb and pb + c or a (by equation 4): so

that d is a measure of a and b.

Also it is their greatest common measure. For, if not, let D, which is greater than d, be their greatest common measure : then because D divides both a and b ; it divides a - pb or c (by equation 1); and therefore b-qc or d (by equation 2); so that D divides d, which is absurd.

Hence d is the greatest common measure required.

COR. Every other common measure of a and b is a measure of d. This is evident from the latter part of the demonstration : for if D be any common measure of a and b, it is proved to divide d.

This proposition is Prop. 2, B. VII., of Euclid's Elements, and, probably on that account, has always been a favourite with examiners.

123. PROP. To find the greatest common measure of three or more numbers.

Let a, b, c, be three numbers, of which the greatest common measure is required.

Take d the greatest common measure of a and b ; and D

the greatest common measure of d and c. common measure required.

D is the greatest

For D divides d, therefore it divides a and b which are multiples of d. Hence D is a common measure of a, b and c. Also it is their greatest common measure.

For, if not, let D' be their greatest common measure. Then, because D' divides a and b, it divides d (122 Cor.).

.. D' measures d and c; and, consequently, it measures D (122 Cor.), which is absurd.

Therefore D is the greatest common measure of a, b and c.

124. PROP. To find the greatest common measure of two algebraic quantities.

The process which we have employed in finding the greatest common measure of numbers requires modification to render it applicable here. A small number being made the divisor of a larger, the quotient and remainder are always integral. With algebraic quantities, however, the case is different. The two quantities ab — b2 and a2 ab obviously have the common measure a-b; yet, if we attempt to divide either by the other, the quotient will be fractional. But a fractional quotient is, from

the nature of the case, inadmissible.

There are two ways in which this introduction of a fractional quotient is evaded in practice: the one by the rejection of a factor in one or more of the divisors; the other by the introduction of a factor into the corresponding dividend.

Now the rejection or introduction of a factor, provided that factor contain no part of the greatest common measure, will produce no effect in the result of finding that measure. For we have seen that every common measure of a and b divides every divisor and dividend a, b, c, etc. If, then, a factor which is no part of the common measure be rejected or introduced into a, or b, or c, etc., it will not affect the divisibility of the quantity in which it is rejected or introduced by d or D, and the argument will remain unchanged.

125. Ex. 1. Find the greatest common measure of 145 and 170.

145)170(1
145

25)145(5
125

20)25(1

5)20(4

hence 5 (the last divisor) is the greatest common measure required.

2. Find the greatest common measure of x2 + 2x + 1 and x2 + 2x2 + 2x + 1.

x2+2x+1)x+2x2 + 2x + 1(x

x2 + 2x2 + x

x + 1(x2 + 2x + 1(x + 1

x2 + x

x + 1 x+1