17. Find the least common multiple of 63, 12, 84, and 7. 18. Find the least common multiple of 54, 81, 63, and 14. 19. Find the least common multiple of 72, 120, 180, 24, and 36.

177.a. The least common multiple of two or more numbers may also be found in the following manner.

First find the greatest common divisor of two of the given numbers; by this divide one of these two numbers, and multiply the quotient by the other. Then perform a similar operation on the product and another of the given numbers; thus continue the process until all of the given numbers have been employed, and the final result will be the least common multiple required.

20. What is the least common multiple of 24, 16, and 12?

Solution. By inspection, we find the greatest common divisor of 24 and 16, is 8. Now 24-8=3; and 3X16=48. Again, the greatest common divisor of 48 and 12, is 12. Now 48÷12 -4; and 4X12-48. Ans:

PROOF.-Resolving the given numbers into their prime factors, 24=2×2×2×3; 16=2×2×2×2; and 12=2×2×3; (Art. 165;) consequently, 2×2×2×2×3=48, the least common multiple. (Art. 175. Obs.)

Oвs. The reason of this rule depends upon the principle, that if the product of any two numbers be divided by any factor which is common to both, the quotient will be a common multiple of the two numbers. Thus, if 48, the product of 6 and 8, be divided by 2, a factor of both, the quotient 24, will be a multiple of each, since it may be regarded either as 8 multiplied by the quotient of 6 by the factor 2, or as 6 multiplied by the quotient of 8 by the same factor. Hence, it is obvious, that the greater the common measure is, the less will be the multiple; and, consequently, the greatest common measure will oduce the least common multiple.

When the common multiple of the first two numbers is found, it is evident, at any number which is a common multiple of it and the third number, will be a multiple of the first, second, and third numbers.

21. What is the least common multiple of 75, 120, and 300? 22. What is the least common multiple of 96, 144, and 720 ? 23. What is the least common multiple of 256, 512, and 1728 ? 24. What is the least common multiple of 375, 85), and 3400 ?

SECTION VII.

FRACTIONS.

ART. 178. When a number or thing is divided into two equal parts, one of those parts is called one half. If the number or thing is divided into three equal parts, one of the parts is called one third; if it is divided into four equal parts, one of the parts is called one fourth, or one quarter; and, universally,

When a number or thing is divided into equal parts, the parts take their name from the number of parts into which the thing or number is divided.

179. The value of one of these equal parts manifestly depends upon the number of parts into which the given number or thing is divided. Thus, if an orange is successively divided into 2, 3, 4,, 6, &c., equal parts, the thirds will be less than the halves; the fourths, than the thirds; the fifths, than the fourths, &c.

OBS. A half of any number is equal to as many units, as 2 is contained times in that number; a third of a number is equal to as many, as 3 is contained times in the given number; a fourth is equal to as many, as 4 is contained in the number, &c.

180. When a number or thing is divided into equal parts, these parts are called FRACTIONS.

OBS. Fractions are used to express parts of a collection of things, as well as of a single thing; or parts of any number of units, as well as of one unit. Thus, we speak of of six oranges; & of 75, &c. In this case the collection, or number to be divided into equal parts, is regarded as a whole.

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181. Fractions are divided into two classes, Common and Decimal. For the illustration of Decimal Fractions, see Section IX.

QUEST.-178. What is meant by one half? What is meant by one third? What is meant by a fourth? What is meant by fifths? By sixths? How many sevenths make a whole one? How many tenths? What is meant by twentieths? By hundreds? When a number or thing is divided into equal parts, from what do the parts take their name? 179. Upon what does the value of one of these equal parts depend? 180. What are frac tions? 181. Into how many classes are fractions divided?

182. Common Fractions are expressed by two numbers, one placed over the other, with a line between them. One half is written thus; one third, ; one fourth, ; nine tenths, &; 1 thirteen forty-fifths, 13, &c.

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The number below the line is called the denominator, and shows into how many parts the number or thing is divided.

The number above the line is called the numerator, and shows how many parts are expressed by the fraction. Thus, in the frac tion, the denominator 3, shows that the number is divided into three equal parts; the numerator 2, shows that two of those parts are expressed by the fraction.

The denominator and numerator together are called the terms of the fraction.

OBS. 1. The term fraction, is of Latin origin, and signifies broken, or sepa rated into parts. Hence, fractions are sometimes called broken numbers.

2. Common fractions are often called vulgar fractions. This term, however, is very properly falling into disuse.

3. The number below the line is called the denominator, because it gives the name or denomination to the fraction; as, halves, thirds, fifths, &c.

The number above the line is called the numerator, because it numbers the parts, or shows how many parts are expressed by the fraction.

183. A proper fraction is a fraction whose numerator is less than its denominator; as, , †, 3/2, 4.

An improper fraction is one whose numerator is equal to, or greater than its denominator; as, ž, Z.

A mixed number is a whole number and a fraction expressed together; as, 4%, 2511.

A simple fraction is a fraction which has but one numerator and one denominator, and may be proper, or improper; as, §, 2.

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A compound fraction is a fraction of a fraction; as, & of 4 of 7, of of 1 of 37.

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QUEST.-182. How are common fractions expressed? What is the number below the line called? What does it show? What is the number above the line called? What does it show? What are the denominator and numerator, taken together, called? Obs. What is the meaning of the term fraction? What are common fractions sometimes called? Why is the lower number called the denominator? Why is the upper one called the numerator? 183. What is a proper fraction? An improper fraction? A Mixed number? A simple fraction? A compound fraction?

A complex fraction is one which has a fraction in its numerator 2층 4 21 5 83

or denominator, or in both; as,

184. Fractions, it will be seen both from the definition and the mode of expressing them, arise from division, and may be treated as expressions of unexecuted division. The numerator answers to the dividend, and the denominator to the divisor. (Arts, 25, 182.) Hence,

185. The value of a fraction is the quotient of the numerator divided by the denominator. Thus, the value of § is two; of 4 is one; of 1 is one third, &c. Hence,

186. If the denominator remains the same, multiplying the numerator by any number, multiplies the value of the fraction by that number. For, since the numerator and denominator answe to the dividend and divisor, multiplying the numerator is the same as multiplying the dividend. But multiplying the dividend, we have seen, multiplies the quotient, (Art. 141,) which is the same as the value of the fraction. (Art. 185.) Thus, the value of §=2; now, multiplying the numerator by 3, the fraction becomes 12, whose value is 6, and is the same as 2X 3.

187. Dividing the numerator by any number, divides the value of the fraction by that number. For, dividing the dividend, divides the quotient. (Art. 142.) Thus, &=2; now dividing the numerator by 2, the fraction becomes, whose value is 1, and is the same as 22.

Hence,

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OBS. With a given denominator, the greater the numerator, the greater will De the value of the fraction.

188. If the numerator remains the same, multiplying the denominator by any number, divides the value of the fraction by that number. For, multiplying the divisor, we have seen, divides th

QUEST.-What is a complex fraction? 184. From what do fractions arise? 185. That is the value of a fraction? 186. What is the effect of multiplying the numerator, while the denominator remains the same? Explain the reason. 187. What is the effect of di viding the numerator? Obs. With a given denominator, what is the effect of increasing the numerator? 188. What is the effect of multiplying the denominator?

quotient. (A t. 143.) Thus, 24-4; now multiplying the denominator by 2, the fraction becomes 24, whose value is 2, and is the same as 42.

189. Dividing the denominator by any number, multiplics the value of the fraction by that number. For, dividing the divisor multiplies the quotient. (Art. 144.) Thus, 2=4; now dividing the denominator by 2, the fraction becomes 24, whose value is 3, nd is the same as 4X2. Hence,

OBS. With a given numerator, the greater the denominator, the less wul de the value of the fraction.

190. It is evident from the preceding articles, that multiplying the numerator by any number, has the same effect on the value of the fraction, as dividing the denominator by that number. (Arts. 186, 189.) And,

Dividing the numerator has the same effect, as multiplying the denominator. (Arts. 187, 188.)

OBS. It will be observed, that multiplying or dividing the numerator of a fraction, has the same effect upon its value, as the same operation has upon a whole number; but, the effect of multiplying or dividing the denominator is exactly contrary to that of the same operation upon a whole number.

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191. If the numerator and denominator are both multiplied or both divided by the same number, the value of the fraction will not be altered. (Art. 146.) Thus, 23; now if the numerator and denominator are both multiplied by 2, the fraction becomes 24, whose value is 3. If both terms are divided by 2, the fracion becomes, whose value is 3; that is, 12—24—5—3.

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192. Since the value of a fraction is the quotient of the numerator divided by the denominator, it follows,

If the numerator and denominator are equal, the value is a unit or one. Thus, -1, 7=1, &c.

QUEST.-189. What is the effect of dividing the denominator? Why? Obs. With a given numerator, what is the effect of increasing the denominator? 190. What may be done to the denominator to produce the same effect on the value of the fraction, as multiplying the numerator by any given number? What, to produce the same effect as dividing the numerator by any given number? 191. What is the effect if the numerator and denominator are both multiplied, or both divided by the same number? 192 When the numerator and denominator are equal, what is the value of the fraction?