Rule I.-Separate the numbers into prime factors; then multiply together ONLY such of those factors, as are necessary to form a product that will contain all the prime factors in each number, using no factor ofterer than it occurs in any one number. NOTE. The solution of Ex. 2, shows that the same factor must be taken the greatest number of times it occurs in either number. After factoring, cancel (cross out) the needless factors. + FIND THE LEAST COMMON MULTIPLE OF 5. 6, 8, 9. 6. 6, 15, 35. 7. 10, 12, 15. Ans. 72. 8. 9, 15, 18, 24. Ans. 360. SECOND METHOD. ART. 120. The L. C. M. of two or more numbers, contains all the prime factors of each of the numbers once, and no other factors. For, if it did not contain all the prime factors of any number, it would not be divisible by that number; and, if it contained any prime factor not found in either of the numbers, it would not be the least common multiple. Thus, the L. C. M. of 4 (2X2), and 6 (2×3), must contain the factors 2, 2, 3, and no others. 1. Find the L. C. M. of 6, 9, and 12. SOLUTION.-Arranging the num bers as in the margin, we find that 2 is a prime factor common to two of them. OPERATION. 2)6 9 12 3)3 9 6 1 3 2 Hence, 2 must be a factor of the L. C. M.; therefore, place it on the left, and cancel it in the numbers of which it is a factor, by dividing by it. 2X 3X 3X 2 = 36 Ans. Next, observe that 3 is a factor common to the quotients and the remaining number, and hence, (Art. 111,) is a factor of the given numbers, and must be a factor of the L. C. Mi therefore, place it on the left, and cancel it in each of the numbers in the 2d line, by dividing by it. As the numbers 3 and 2, in the 3d line, have no common factors to cancel, we do not divide them. Thus we find, that 2, 3, 3, and 2, are all the prime factors in the given numbers; hence, their product, 2×3×3X2=36, is the L. C. M. of 6, 9, and 12. In this operation, let the learner notice, 1st. The number 36 is a common multiple, because it contains all the prime factors in each of the numbers; it is the least C. M., because all the needless factors were canceled by dividing. 2d. To cancel needless factors, divide by a prime number. By dividing by a composite number, in some cases, all the needless factors are not canceled; thus, in the preceding example, 6 will exactly divide two of the numbers; but, In dividing by 6, a factor, 3, is left uncanceled in the multiple 9, and thus the L. C. M. is not obtained. 2. Find the L. C. M. of 6 and 10. 3. Of 15, 21, 35. Ans. 30. Rule II-1. Place the numbers in a line, divide by any prime number that will divide two or more of them without a remainder, and place the quotients and undivided numbers in a line beneath. 2. Divide this line as before: continue to divide till no number greater than 1 will exactly divide two or more of the numbers. 3. Multiply together the divisors and the numbers in the lowest line, and their product will be the least common multiple. REM. If the given numbers contain no common factor, their product will be the L. C. M. Thus, the L. C. M. of 4, 5, and 9, is 4X5X9180. REVIEW. 119. When is one number divisible by another? Give an example. When not divisible? Give an example. What factors must 'the Com. Mul. contain? 119. What prime factors must the L. C. M. not L. C. M. of 6, 10, and 18, and explain the operation. contain? Find the What is Rr le I? 119. NOTE. How often must the same factor be found in the L. C. M? XI. COMMON FRACTIONS. ART. 121. A single thing, (Art. 1), is called a unit, or one, which be divided into equal parts. may Thus, suppose 3 apples are to be equally divided between 2 boys: after giving one to each, there would remain one to be divided into two equal parts, to complete the division. The equal parts into which a unit is divided are fractions. ART. 122. When a unit, or single thing, is divided into two equal parts, one of the parts is one-half. If it is divided into three equal parts, one of the parts is one-third; two of the parts, two-thirds. If divided into four equal parts, one of the parts is one-fourth; two of the parts, two-fourths; and three of the parts, three-fourths. If divided into five equal parts, the parts are fifths; if into six equal parts, sixths, and so on. Hence, When a unit is divided into equal parts, the parts are named from the number of parts into which the unit is divided. REVIEW.-120. Ex. 1. Why divide by 2? By 3? Why multiply together the numbers 2, 3, 3, and 2? Why is 36 a Com. Mul. of 6, 9, and 12? Why the least? To cancel needless factors, why not divide by. a composite number? What is Rule II? 120. REM. If the numbers contain no common factor, how is their L. C. M. found? 121. How do you divide 3 apples equally between 2 boys? When a unit is divided into equal parts, what are the parts called? ART. 123. The value of one of the parts depends on the number of parts into which the unit is divided. Thus, if 3 apples of equal size be divided, one into 2, another into 3, and another into 4 equal parts, the thirds will be less than the halves, the fourths less than the thirds. ART. 124. Fractions are divided into two classes, Common and Decimal. Common Fractions are expressed by two numbers, one above the other, with a horizontal line between them. Thus, one-half is expressed by; two-thirds by 3. The number below the line is the denominator: it denominates, or gives name to the fraction. It shows the number of parts into which the unit is divided. The number above the line is the numerator: it numbers the parts, showing how many parts are taken. Thus, in the fraction, the denominator, 5, shows that the unit is divided into five equal parts, and the numerator, 3, shows that the fraction contains 3 of those parts. The numerator and denominator together, are called the terms of the fraction. Thus, the terms of g, are 3 and 5. ART. 125. ANOTHER METHOD. In the definition of numerator and denominator, reference is had to a unit only. This is the simplest method of considering a fraction; but, there is another mode : ILLUSTRATION. To divide 2 apples equally among 3 boys, divide each apple into three equal parts, making 6 parts in all; then give to each boy 2 of the parts, expressed by . REVIEW.-122. When a unit is divided into two equal parts, what is one part called? When divided into three equal parts, what is one part called? What two parts? When divided into four equal parts, what is one part called? Two parts? Three parts? When a unit is divided into equal parts, from what are the parts named? 123. On what does the value of one of the parts depend? Which is greater, 1-half or 1-third? 1-third cr 1-fourth? 1-fourth or 1-fifth? 124. Into what two classes are fractions divided? How are common fractions expressed? What is the number below the line? Why? What the number above the line? Why? What are the terms? In selecting one boy's share, take the 2 parts from one apple, or 1 part from each of the two apples; hence, expresses either 2 thirds of 1 thing, or 1 third of 2 things. Also, expresses 3 fifths of 1 thing, or 1 fifth of 3 things. Therefore, the numerator of a fraction expresses the number of units to be divided; and the denominator the divisor, or what part is taken from each. Hence, A fraction is expressed in the form of an unexecuted division, is the quotient of 1 (numerator) ÷ 3 (denom.); ART. 128. Since fractions arise from division, one of the signs of division (Art. 40,) is used in expressing them; the numerator being written above, and the denominator below, a horizontal line. Thus, three-fifths is written ; two-sixths is written . TO READ COMMON FRACTIONS. Read the number of parts taken, as expressed by the numerator; then the size of the parts, as expressed by the denominator. TO WRITE COMMON FRACTIONS. Write the numerator, place a horizontal line below it, under which write the denominator. REM.—In reading, means two-thirds of one. There are two other methods, (Art. 125): thus, may be read, cne-third of two, or two divided by three; but these methods are rarely used. REVIEW.-125. What do two-thirds express? What does the numerator of a fraction express? The denominator? In what form is a fraction expressed? What is the dividend? The divisor? The quotient? Give examples. 126. What sign is used in fractions? Why? Ilow is a common fraction read? Give examples. How written? Give examples. |