RULE.-Divide the greater number by the less, and if there is a remainder, divide the divisor by it, and continue dividing the last divisor by the last remainder, until nothing remains. The last divisor will be the greatest common divisor. If more than two numbers are given, find the greatest common divisor of two of them, then of this divisor and one of the other numbers, and so on, to the last. The last divisor will be the greatest Reduce the following fractions to their lowest terms by finding the greatest common divisor of their terms and then dividing both numerator and denominator by the G. C. D.: 16. A farmer has three tracts of land containing 240, 320, and 440 acres, respectively, which he wishes to divide into the largest possible fields of equal size. How many acres will there be in each field, and how many fields in each tract? LEAST COMMON MULTIPLE. 136. A Multiple of a number is any number that will exactly contain it. Thus, 15 is a multiple of 5; 63 is a multiple of 9; and 80 is a multiple of 16. 137. A Common Multiple of two or more numbers is any number that will exactly contain each of them. Thus, 48 is a common multiple of 6, 8, and 12; and 60 is a common multiple of 5, 10, and 20. 138. The Least Common Multiple of two or more numbers is the least number that will exactly contain each of them. Thus, 12 is the least common multiple of 3, 4, and 6; and 40 is the least common multiple of 5, 8, and 10. The initials L. C. M. may be used for least common multiple. 139. PRINCIPLE.-The least common multiple of two or more numbers is equal to the product of all the different prime factors used the greatest number of times they occur in any one number. WRITTEN EXERCISES. 140. 1. Find the least common multiple of 12, 14, and 20. 12 2 X 2 X 3 2425 14 = 2 X 7 20=2 X 2 X 5 L. C. M. = 2 × 2 × 3 × 5 × 7 = EXPLANATION.-The numbers are first separated into prime factors. The factor 2 420 occurs the greatest number of times but twice in any one num ber, so that it must appear twice as factors of the least common multiple. The factors 3, 5, and 7 are the other different factors, and they appear but once in any number, hence each of them must appear but once as factors of the least common multiple. Therefore the product of 2, 2, 3, 5, and 7, or 420, is the least common multiple of the numbers. common multiple of the numbers may also be found by placing them in a horizontal line 3 L. C. M. = = 2 × 2 × 3 × 7 X 5420 and dividing them by any prime number that will exactly divide at least two of them, and thus continue until quotients are found that are prime to one another. The product of the divisors and quotients will be the least common multiple. In finding the least common multiples of numbers, all numbers that are factors of other given numbers may be disregarded. Thus, the least common multiple of 12, 15, 36, and 90 is the same as the least common multiple of 36 and 90; for 12 is a factor of 36, and 15 is a factor of 90. Find the least common multiple of the denominators of the following fractions: 16. What is the smallest piece of land that can be divided into fields of 16, 18, and 20 acres each? 141. In the operations of practical arithmetic it is seldom necessary to find the least common multiple of numbers greater than 109. In such cases the factors composing the least common multiple may be obtained by inspection. When numbers cannot be readily factored by inspection, the factors may be found by the method of finding the greatest common divisor of such numbers (Art. 131). 1. Find the least common multiple of 345 and 391. 345) 391 (1 46) 345 (7 = G. C. D. 23) 46 (2 46 391 17 EXPLANATION.-Since the factors can not be obtained easily by inspection, the G. C. D. is found, which is 23. Dividing the given numbers by 23 the quotients 15 and 17 are obtained, which factors are all prime to one another. Hence, the product of 23, 15, and 17, or 23 X 15 X 17 =5865 5865, is the L. C. M. RULE.-Find the greatest common divisor of the numbers and divide each of the given numbers by it. Then take the product of the greatest common divisor and the quotients. The result will be the least common multiple. Find the least common multiple of: 11. A can walk around a certain island in 16 hrs., B in 18 hrs., and C in 20 hrs. If they start together and keep walking each at his own rate, how many days will elapse before they are all three together at the starting point, and how many times will each have gone around the island? 12. The periods of three planets which move uniformly in similar orbits around the sun, are respectively 275, 325, and 365 days. Supposing all three in conjunction at a given moment, in how many days will they be in conjunction again? FRACTIONS. 142. A Fraction is one or more of the equal parts of anything. A fraction is expressed by two numbers, written one above the other with a line between them. Thus, is a fraction. 143. The Denominator of a fraction shows into how many parts a thing has been divided. It is written below the line. Thus, in the fraction, 4 is the denominator, and shows that something has been divided into 4 equal parts. 144. The Numerator of a fraction shows how many parts form the fraction. It is written above the line. Thus, in the fraction, 3 is the numerator. It shows that the fraction contains 3 of the 4 equal parts. 145. The Terms of a Fraction are the numerator and denominator together. 146. A Proper Fraction is one whose numerator is less than its denominator. Thus,,, and 13 are proper fractions. The value of a proper fraction is less than 1. 147. An Improper Fraction is one whose numerator equals or exceeds its denominator. Thus,,,, and 32 are improper fractions. The value of an improper fraction is 1 or more than 1. 148. A Mixed Number is a number expressed by an integer and a fraction. Thus, 3, 73, and 18 are mixed numbers. 960, 149. The Unit of the Fraction is the unit which is divided into equal parts. 1. A Common Fraction is one whose unit has been divided into any number of equal parts. 2. A Decimal Fraction is one whose unit has been divided into tenths, hundreds, thousands, etc. 150. A Fractional Unit is one of the equal parts into which a unit has been divided. 151. An Integer may be expressed fractionally by writing 1 under it as a denominator. 152. The Reciprocal of a number is 1 divided by that number. Thus, the reciprocal of 4 is 1. 1. A fraction may be regarded as a number of equal parts of a unit, a number of equal parts of one thing, or one equal part of a number of things. Thus, three fourths may be regarded as three fourths of one, three fourths of one thing, or one fourth of three things. 2. A fraction also expresses unerecuted division. 153. Fractions are read by naming first the number of fractional units, and then the kind. Thus, is read five ninths; is read six forty-firsts. Copy, read, and tell the kind of fractions: 1. Five sixths. Six ninths. Eight elevenths. 2. Three tenths. Twelve elevenths. Four sevenths. 3. Eight twentieths. Seventeen nineteenths. 4. Nine and two thirds. Five and three fifths. 5. Sixty-two forty-eighths. Thirteen twenty-fifths. 154. To Analyze a fraction is to explain what is expressed by it. 1. Analyze the fraction *ANATION.— represents 3 of 4 equal parts of a thing; or it may fourth of 3 or 3 divided by 4. It is read three fourths. |