Exercises in the division of general expressions of quantity:

1st. Divide a by a?

...................... Ans. a3. ...Ans. aTM-3. .......Ans. be-'. .........Ans. a. ....Ans. 1.

.Ans. ab+ac+be.

Ans. axTM-1+bxTM-2+cx®-3+nx+p+ 1;

Verifications of Multiplication and Division.

101. The product of two factors having been found, the accuracy of the result may be tested by making the multiplicand multiplier, and the multiplier multiplicand, and repeating the multiplication. Then, if both products agree, it is to be presumed that the result is accurate; for, as the combinations effected are different, it is unlikely that the same error can occur in both calculations.

Another verification of the accuracy of a product is supplied by division, for the product of the divisor and quotient is the dividend. Whence, if the product of two factors is made dividend, and one of the factors divisor, the quotient must be equal to the other factor.

As the process of division is more complicated than that of multiplication, this verification is liable to the objection of Article 44.

102. A convenient, though not an indispensable, preliminary to the verification of a quotient is the subtraction of the remainder (if there is one) from the dividend; the division is thus rendered exact.

Then, since the dividend is equal to the product of the divisor and quotient, if the divisor is multiplied by the quotient, and the product resulting from this multiplication is equal to the dividend (or the dividend less the remainder), it is to be presumed that the division has been accurately performed.

Another verification, without subtraction of the remainder, may be obtained by making the quotient divisor, and repeating the division. Then the number which before was divisor ought to be quotient now, and the remainder the same as before.

Another verification of multiplication and division is given, Article 131.

DIVISIBILITY OF

MEASURE AND

SECTION VI.

NUMBERS.

GREATEST COMMON LEAST COMMON MULTIPLE OF TWO OR MORE NUMBERS.

103. A tentative method of decomposing a large number into factors has been explained in a preceding Article. The possibility of exactly dividing one number by another can always be determined by trial, and it is often the easiest way of ascertaining whether the division can be effected or not.

There are, however, tests by which the divisibility of numbers by certain divisors can be readily ascertained. Some of the most simple and elementary shall be given in the course of this Section.

As the research of the divisors of one number, the common divisors of two numbers, and the least common multiple of two or more numbers, depends on a common principle, it seems useful to take these subjects in connection with each other. Some new terms are necessary, of which the following are definitions:

a. One whole number is said to be exactly divisible by another whole number when there exists a third whole number, which, multiplied by the second, produces the first.

b. Every whole number which exactly divides another whole number is called a factor, divisor, measure, sub-multiple, or aliquot part of that number.

c. Every whole number which has no divisor, excepting unity, is called an absolute prime number, or, simply, a prime number.

d. Every whole number which exactly divides each of two other whole numbers is called a common divisor or common measure of these numbers. e. Two whole numbers are said to be prime to each other when they have no common measure but unity.

f. From definitions c, e, it follows that a prime number, which is not a sub-multiple of another whole number, is prime with that other number, for they can have no common measure but unity.

g. Two whole numbers not prime to each other may have several common divisors. Taking, for example, 12 and 18, the numbers 2, 3, 6, are divisors common to both. Of these common divisors, or common measures, if there are several, one must be greater than the others. This is termed the Greatest Common Measure; which expression, for brevity, is sometimes written, g. c. m.

h. A whole number which is exactly divisible by another whole number is called a multiple of that by which it is divided; and a whole number which is exactly divisible by two or more numbers, a common multiple of these numbers.

i. The same divisors have many common multiples. For example, 12, 24, 36, . . . . are all common multiples of the numbers 2, 3, 4. Of these common multiples one must be the least of those which are exactly divisible by the given divisors; this is called their Least Common Multiple. The expression, least common multiple, is sometimes written in the abridged form, 1. c. m.

k. To avoid prolixity of expression and ambiguity, the following abbreviations are also employed; viz.

N, N', N" d, d', d" P, P, P

7, 7, 7'

m, m', m'

for numbers (N' is read N. one, &c.)

for divisors or measures.

for the parts into which a number is divided.
for remainders.

for quotients or factors of a number.

.. for coefficients of a number.

104. Every number which measures another number measures also any multiple of that other number.

Let N be a multiple of d; mN, any multiple of N, is also a multiple of d.

Since N is a multiple of d, let N=qd;

Whence mN=mqd=mqxd; that is, if d is contained q times in N, it is contained mq times in mN. Consequently, as m, q, are by hypothesis whole numbers, the proposition is established.

105. Every number which is decomposed into two parts, each divisible by a second number, is itself divisible by this second number.

Let a number, N, be divided into the parts P, P'; and let P, P' be both divisible by d; N shall be divisible by d.

Since P is divisible by d, let P=qd.

P'

d,,, P=q'd.

Therefore P+P=qd+q'd=(q+q)d (Art. 70).

But P+P N; therefore N=(q+g')d.

Now q, q' are, by hypothesis, whole numbers; therefore the sum of I, I is a whole number, and consequently d is a measure of N.

106. If a number is decomposed into two parts, every number which measures the whole and one of the parts must also measure the other part.

Let N=P+P', and let N, P be each divisible by d; P′ shall be divisible by d.

By hypothesis, N=qd (q being a whole number),

and P=q'd (q being also a whole number);

let P'q'd (q" being any quotient, whole or not); then, since N=P+P',

qd=q'd+q'd=(q' +q′′)d.

Now 9 is a whole number; therefore the sum of q, q'"' is a whole number.

But is also a whole number; therefore q" must be a whole number; otherwise one whole number must be equal to another whole number and a fraction, which is absurd.

Therefore q' is a whole number, and consequently P' is divisible by d.

107. The preceding principles are sufficient for the investigation of the greatest common measure of two numbers.

To apply them, let it be required to find the greatest common measure of the numbers 637 and 143.

This greatest common measure cannot exceed the less number 143; and seeing that 143 divides itself, if it also divides 637, it must be the greatest common measure sought.

Dividing 637 by 143, the quotient is 4, and the remainder 65.

Or, 637-143 × 4+65.

measure.

Wherefore 143 is not the greatest common

The greatest common measure of the numbers 637 and 143 is therefore less than 143.

Every number which divides 143 divides also 143 × 4, or 572 (Art. 104); and since the greatest common measure divides 637, it must also divide 637-143 × 4 or 65 (Art. 106). Whence the greatest common measure of 637 and 143 cannot be greater than that of 143 and 65.

Again, the greatest common measure of 143 and 65 dividing the two parts of 637 (viz. 4× 143 and 65) must divide the whole number 637 (Art. 105); and being an exact divisor of 637 and 143, it cannot be greater than the greatest common measure of 637 and 143. Whence, since the greatest common measure of 637 and 143 is not greater than that of 143 and 65, and the greatest common measure of 143 and 65 is not greater than that of 637 and 143, it follows that they are equal.

Following with 143 and 65 the process and reasoning employed with 637 and 143, it is found that 143÷65=2+13_remainder, and that the greatest common measure of 143 and 65 must be the same as that of 65 and 13.

The question is therefore reduced to the investigation of the greatest

common measure of 65 and 13.

Dividing 65 by 13, an exact quotient is found, namely 5.

Whence, as 13 divides itself and 65, it is a common measure of 65 and 13.

Being also the greatest number by which 13 can be divided, it is the greatest common measure of 65 and 13.

But it has been shown that the greatest common measure of 65 and 13 is also the greatest common measure of 143 and 65, and that the greatest common measure of 143 and 65 is the greatest common measure of 637 and 143.

Therefore 13 is the greatest common measure of 637 and 143.

The calculation may be made as under :

143)637(4

572

65)143(2
130

13)65(5

65

0

108. The following practical rule is deduced from Article 107.

To find the greatest common measure of two numbers, Divide the greater number by the less; if there is no remainder, the less number is the greatest common measure of the two numbers.

If there is a remainder, divide the less number by this remainder. If the division is exact, the first remainder is the greatest common

measure.

But if the second division gives a remainder, divide the first remainder by the second. If this division is exact, the second remainder is the greatest common measure: if not, and so long as the division is not exact, continue to divide the last remainder but one by the last.

An exact quotient being obtained, the last divisor is the greatest common measure of the two given numbers.

If the last divisor is unity, the two numbers, having no common measure but unity, are prime to each other.

109. If the process for finding the greatest common measure be employed upon two numbers which are prime to each other, the last remainder found must be unity; for, from the nature of the process, each remainder is less than that which preceded it; and a remainder equal to zero cannot be obtained before a remainder equal to 1, for the divisor greater than 1 which gives a remainder =0, must be a common measure of the numbers; that is, of two numbers prime to each other, which is absurd.

Whence, in the case of two numbers prime to each other after more or fewer divisions, a remainder equal to unity must always be found.

110. When any remainder is a prime number, and not a measure of the preceding remainder, a remainder equal to 1 must be found (Art. 103 f), and the numbers proposed have no greatest common measure.

Note.-Two numbers may have only one common measure greater than unity. This is called their greatest common measure.

111. Every number which measures the product of two factors, and which is prime with one factor, must measure the other factor.

Let NN'=md (m representing a whole number), and let N and d be prime to each other; then N ́ is measured by d.

To establish this principle, let it be supposed, first, that N* is greater than d, and let the process for finding the greatest common measure of N and d be employed.

Since N and d are, by hypothesis, prime to each other, this process must end in a remainder =1 (Art. 109).

The detail is contained in the first column of the subjoined table.

In a second column the equal quantities of the first are multiplied by the other factor N'. The products are equal; for if equal quantities are multiplied by the same quantity (or have the same quantity added to them) the results must be equal.

* If N is less than d, the application of the greatest common measure is made by dividing d by N; in other respects the process does not differ from that given.

In a third column the equal quantities of the second are divided by the measure d. The results again are equal, for if equal quantities are divided by the same quantity (or have the same quantity subtracted from them) the results are equal.

N=qd +r N'N N'qd +N'r

d=q'r +r' N'd N'gr +N'r

r =q'r'+r" N'r =N'q'r'+N ́r”

r=q"r"+r"" | N'r'=N'q""r"+N'r""

N'N

d=N'q
N'g'r

N'r

+d

N'r

N' =

+ d

d

Considering now the quantities in the third column;

N'N

d

is a whole number, by hypothesis.

N'q is a whole number; since N' and q are both whole numbers, there

N'r
d

therefore (Art. 106) is a whole number.

In the same manner it may be shown that the quantities are whole numbers.

And since in the succession of remainders, r, r', r'',

der=1 must be found, (Art. 109); it follows that and, by consequence, that N' is measured by d.

N'x1
d

is a whole number,

112. Every absolute prime number which measures the product of two factors must measure one of the factors.

Let d be an absolute prime number which divides the product NN'; d must divide either N or N'.

For if d does not divide N it is necessarily prime with that factor (Art. 103 e), and therefore it must divide the other factor N' (Art. 111).

113. Every absolute prime number which divides N2, or in general any power, N", of N must divide N.

For N2 NxN. Now, every absolute prime number which measures NXN must divide one of the factors (Art. 112). For the same reason every absolute prime number which divides N3 must divide N or N2; and to divide N2, it must divide N.

This proof may be extended to any power of N.

114. Every number which is prime with both the factors of a product is also prime with the product.

Let d be prime to each of the numbers N, N'; d shall be prime to the product N, N'.

For any absolute prime number which divides d and N N' must divide either N or N' (Art. 112). Whence d and N, or d and N', cannot be prime to each other, which is contrary to the hypothesis.

115. The product of many factors, N, N', N".... can contain no prime factors which are not contained in some of the numbers N, N', N". . .*

For every prime number which divides N N' N", but does not divide N", must divide N N' (Art. 112), and every prime number which divides N N' must divide N or N'.