If, then, the DIVISOR remains the same, MULTIPLYING THE DIVIDEND BY ANY NUMBER, MULTIPLIES THE QUOTIENT BY THE SAME

NUMBER.

If a divisor is contained in a dividend a certain number of times, the same divisor is contained in half that dividend one half as often; in one third of that dividend, one third as often, and so on.

If, then, the DIVISOR remains the same, DIVIDING the divideND EY ANY NUMBER, DIVIDES THE QUOTIENT BY THE SAME NUMBER. If a divisor is contained in a dividend, a certain number of times, twice that divisor is contained in the same dividend, one half as often ; 3 times that divisor is contained in the same dividend, one third as often, and so on.

If, then, the DIVIDEND remains the same, MULTIPLYING THE DIVISOR BY ANY NUMBER, DIVIDES THE QUOTIENT BY THE SAME NUMBER.

If a divisor is contained in a dividend, a certain number of times, half that divisor is contained in the same dividend, twice as often; one third of that divi or, 3 times as often, and so on.

If, then, the DIVIDEND remains the same, DIVIDING THE DIVISOR BY ANY NUMBER, MULTIPLIES THE QUOTIENT BY THE SAME NUM

BER.

It seems then, that multiplying the dividend has the same effect on the quotient as dividing the divisor; and that dividing the dividend has the same effect as multiplying the divisor.

Of course, multiplying both divisor and dividend produces opposite effects upon the quotient, and dividing both divisor and dividend produces opposite effects upon the quotient. Then,

IF THE DIVISOR AND DIVIDEND BE BOTH MULTIPLIED, OR BOTH DIVIDED BY THE SAME NUMBER THE QUOTIENT WILL NOT BE ALTERED.

Now as every Fraction is an instance of division, in which the numerator is the dividend, the denominator the divisor, and the value of the Fraction, the quotient, we may apply the above principles to them. Of course,

MULTIPLYING THE NUMERATOR OF A FRACTION, MULTIPLIES THE VALUE, AND DIVIDING THE NUMERATOR DIVIDES THE VALUE. And,

MULTIPLYING THE DENOMINATOR OF A FRACTION DIVIDES THE VALUE, AND DIVIDING THE DENOMINATOR MULTIPLIES THE VALUE. And,

MULTIPLYING THE NUMERATOR PRODUCES THE SAME EFFECT UPON THE VALUE AS DIVIDING THE DENOMINATOR; AND DIVIDING THE NUMERATOR PRODUCES THE SAME EFFECT AS MULTIPLYING THE DENOMINATOR.

And,

MULTIPLYING THE NUMERATOR PRODUCES AN OPPOSITE EFFECT UPON THE VALUE, FROM MULTIPLYING THE DENOMINATOR; AND DIVIDING THE NUMERATOR PRODUCES AN OPPOSITE EFFECT FROM DIVIDING THE DENOMINATOR. Therefore,

IF THE NUMERATOR AND DENOMINATOR BE BOTH MULTIPLIED, OR DIVIDED BY THE SAME NUMBER, THE VALUE OF THE FRACTION WILL NOT BE ALTERED.

MENTAL EXERCISES.

§ XLII. 1. Which Fraction has the greatest value 12 or ? Ans. Neither, because 6 is in 12, twice, and 3 is in 6 twice; or =2, and §=2.

2. Which is the greatest 30 or ? Why?

3. Which is the greatest or?

Why?

4. Which is the greatest or 2 ? Ans. Neither, for 6 is 3 times 2, or 3 twos, and 8 is 4 times 2, or 4 twos. It is plain, that 3 twos is the same part of 4 twos that 3 ones is of 4 ones, or that 3 is of 4.

5. Which is the greatest or 7? Why? or ? Why? org? Why? org? Why? org? Why? We see then, that Fractions of the same value may be expressed by very different numbers.

The numerator and denominator of a Fraction are called TERMS. WHEN A FRACTION IS EXPRESSED BY THE SMALLEST NUMBERS POSSIBLE, IT IS SAID TO BE IN ITS LEAST OR LOWEST TERMS.

When a Fraction is not in its lowest terms, it is plain that to make it so, we must diminish the terms, in such a manner as not to alter the value of the Fraction. This can be done by dividing both, by some number which will divide them, without remainder. (§ XLI.) 1. Reduce the following Fractions to their lowest

2. Reduce the following Fractions to their lowest ; ; ; &; ; ; ; &

terms. 4.5. 21

1.5

50
4
2

9

18 14

16

30 35

8

9

63

3. Reduce the following Fractions to their lowest terms. 17; 11; 11; 82; 77; 42; 24; £#; 14; 46: 74; 28; 13; 12; 18.

First divide by 5, and then by 5 again. 5)}} 5 = 5) 1 4 5 =3, Ans.

5. Reduce; 19; 11; lowest terms. Ans. ; ; ;

The rule then, seems to be—

7 and; to their and.

DIVIDE BOTH TERMS OF THE FRACTION BY ANY NUMBER WHICH WILL DIVIDE THEM WITHOUT A REMAINDER, AND THE QUOTIENTS IN THE SAME MANNER, UNTIL NO NUMBER GREATER THAN 1 will divide both, WITH

OUT A REMAINDER.

The pupil will soon find, that there are some numbers which cannot be divided without remainder, except by themselves or by 1.

Thus, 7 can be divided without remainder by no numbers except 7 and 1. 13, 17, &c. are such numbers.

A NUMBER WHICH CAN ONLY BE DIVIDED BY ITSELF, OR BY 1, IS CALLED A PRIME NUMBER.

Numbers which admit of division, or of being resolved into facLors, we have already seen (§ XII.) are called composite numbers. AN EVEN DIVISOr of a number, that is, one which will divide it without remainder, IS CALLED A MEASURE OF THAT NUmber.

Thus, 3 is a measure of 15, because 3 divides 15 without remainder. 7 is a ineasure of 35, &c.

Every number is a measure or even divisor of itself.

10 may be divided by 2, 5, or 10. Of course, any number of 10s may be divided by 2, 5, or 10. Then, if the right hand figure of a number be a cypher, that number is divisible, that is, it may be divided by 2, 5, or 10. Of course, if the right hand figure of a number be divisible by 2, the whole number is divisi ble by 2. And, if the right hand figure of a number be 5, the whole number is divisible by 5.

100 may be divided by 4. Of course, any number of 100s may be divided by 4. Then, if there be two cyphers at the right of a number, the number is divisi ble by 4. Of course, if 4 will divide two figures on the right of a number, it will divide the whole number.

1,000 may be divided by 8. Of course, any number of 1,000s may be divided by 8. Then, if there be three cyphers at the right of a number, the number is divisible by 8. And, if 8 will divide three figures on the right of a number, it will divide the whole number.

It has been already shown, (§ IX.) that if the sum of the figures, which com. pase a number, be divisible by 9 or by 3, the whole number is divisible by 9 or by 3.

20 is divisible by 4. Of course, any number of times 20 is divisible by 4. Then, if the tens be even, we need only try the right hand figure by 4, to discover whether it will divide the whole number. 200 is divisible by 8. Of course, any number of times 200 is divisible by 8. Then, if the hundreds be even, we need only try the two right hand figures by 8, to discover whether it will divide the whole number.

To discover whether a Prime number will divide, we must make actual trial. 1. Find the measures of the following numbers, omitting the number itself, as being of course a measure. 18, 27, 20, 21, 24, 48, 72. Ans. Of 18, the measures are 2, 3, 6, 9. Of 27; 3, 9. Of 20; 2, 4, 5, 10. Of 21; 3, 7. Of 24; 2, 3, 4, 6, 8, 12. Of 48; 2, 3, 4, 6, 8 12, 16, 24. Of 72; 2, 3, 4, 6, 8, 9, 12, 18, 24, 36.

2. Find the measures of the following numbers, 108, 120, 432, 936, 846.

3. Reduce the following Fractions to their lowest 9720 1728; 12; 11; 1; A48;

terms.

806

112

4.86

1158

;

144

64

3578164

4746 38433

2703 ; 32208476' Ans.; }; }; 7;

1582

4. Reduce the following: #44; 1988:

998811

933 8764 103284

§ XLIII. In almost every one of the above instances the pupil has found it necessary to perform several successive divisions. But

if he could have found at once, the greatest number, which would divide both terms without remainder, only one division would have been necessary.

A NUMBER WHICH IS A MEASURE OF TWO OR MORE NUMBERS IS CALLED THEIR COMMON MEASURE.

THE GREATEST NUMBER WHICH WILL MEASURE TWO OR MORE NUMBERS IS CALLED THEIR GREATEST COMMON MEASURE.

MENTAL EXERCISES.

1. What is the greatest common measure of 4 and 6? of 8 and 12? of 14 and 16? of 6 and 9?..

2. What is the greatest common measure of 4 and 10? of 4 and 14? of 9 and 15? of 10 and 15? of 7 and 21 ?

3. What is the greatest common measure of 3 and 27? of 30 and 9 of 15 and 20? of 20 and 30? of 30 and 40?

4. What is the greatest common measure of 2 and 20? of 8 and 26 of 6 and 16? of 18 and 24?

5. What is the greatest common measure of 12 and 16? of 16 and 20? of 12 and 18? of 16 and 18?

6. What is the greatest common measure of 85 and 28 of 28 and 21? of 21 and 14? of 14 and 2?

In the case of small numbers like the above, there is no difficulty in finding the greatest common measure. The following are not so easy.

7. Find the greatest common measure of 125 and 375. We cannot tell what this is at first sight. But we see by the above examples, that the smaller number is sometimes, itself the measure sought. Let us try whether this is not the case in the present instance.

375

We see by this illustration, that 125 is a measure 125)375(3 of 375, and as 125 cannot be measured by a number greater than itself, it is, of course, the greatest common measure sought.

144

24)144)6 144

8. Find the greatest common measure of 144 and 168. Try first, as before, whether 144 is the measure. 144)168(1 168 144 leaves 24 remainder. Now if 24 will measure 144, it will also measure 144+24=168. 144÷24-6. 24 then is a common measure of 144 and 168. It is also the greatest common measure, for any number which will measure 144 and 168, will measure their difference. For it is contained an even number of times in 144, and also a greater even number of times in 168. But taking an even number of times any thing from an even number of times the same thing, will leave an even number of times that thing. But 24 cannot be measured by any number greater than itself. Therefore, there can be no greater common measure of 144 and 168, than 24. The explanation would be similar, if several successive divisions took place. Therefore, to find the greatest common measure of two numbers,

DIVIDE THE Greater numbER BY THE LESS, THAT DIVISOR BY THE REMAINDER, AND SO ON, ALWAYS DIVIDING THE LAST DIVISOR BY THE

LAST REMAINDER, UNTIL SOME REMAINDER IS LEFT.

THE DIVISOR

WHICH LEAVES NO REMAINDER IS THE GREATEST COMMON MEASURE.

If it be required to find the greatest common measure of several numbers, find first the greatest common measure of two of them, then the greatest com. mon measure of this common measure and a third, then the greatest common measure of this and a fourth, &c. The last common measure, will be that required. Thus,

8. Find the greatest common measure of 12 and 30 and 9.

6 is the greatest common measure of 12 and 30. 3 is the greatest common measure of 6 and 9, and of course of the three given numbers 12, 30 and 9.

11. Find the greatest com. meas. of 72 and 96. A. 14. Of 330 and 462. A. 66. Of 126 and 342. A. 18. Of 84, 28 and 42. A. 14. Of 34, 746, 69,492 and 38,433. A. 3. Of 1,872, 3,456. 98,712 and 531,711. Of 12,572, 92,416, 2,354 and 34,564.

This principle may be applied with advantage to the reducing of Fractions to lower terms. Take the following examples.

The pupil has, no doubt, by this time observed that there are some numbers which have no common measure greater than one. These numbers are said to be PRIME TO EACH OTHER. This definition must not be confounded with that given of prime numbers; for num. bers which are themselves composite, may be prime to each other. Thus 25 and 27, both of which are composite, are prime to each other. A number is prime to itself, which has no measure greater than 1. Numbers are prime to each other, which have no common measure greater than 1.

The

We may here make one more suggestion, which will be useful in finding the measures of numbers by trial. 2 and 5, which are prime to each other, are measures of 20. If 20 be resolved into the factors, 5 and 4, we know that 2, which is a measure of 20, must be a measure of 4, since it is not a measure of 5, and has no factor, which is a measure of 5. 20 may therefore be divided successively by 5 and 2, and, consequently, (§ XXIX.) by their product 10. same may be shown of any two or more numbers, prime to each other, which are measures of a third number. Hence, If a number be divisible by two or more numbers, which are prime to each other, it is likewise divisible by their product. If the numbers are not prime to each other, we cannot make this inference.

MENTAL EXERCISES.

§ XLIV. 1. Change the Fractions and to others having the same denominator. This can easily be done, because we know that is the same as 2.

18

2. Reduce the Fractions and to the same denominator.

3. Reduce and to the same denominator.