SECOND METHOD. 105. 1. What is the least common multiple of 4, 6, 9, and 12? 6..9 12 OPERATION. 2 4 2 2 3 3 3 9 6 3..9.. 3 3 2 × 2 × 3 × 3 = 36, Ans. ANALYSIS. We first write the given numbers in a series, with a vertical line at the left. Since 2 is a factor of some of the given numbers, it must be a factor of the least common multiple sought. Dividing as many of the numbers as are divisible by 2, we write the quotients and the undivided number, 9, in a line underneath. We now perceive that some of the numbers in the second line contain the factor 2; hence the least common multiple must contain another 2, and we again divide by 2, omitting to write down any quotient when it is 1. We next divide by 3 for a like reason, and still again by 3. By this process we have transferred all the factors of each of the numbers to the left of the vertical; and their product, 36, "must be the least common multiple sought, (104, III.) 2. What is the least common multiple of 10, 12, 15, and 75? 2,5 2,3 5 OPERATION. 10..12..15..75 6.. 3..15 ANALYSIS. We readily see that 2 and 5 are among the factors of the given numbers, and must be factors of the least common multiple; hence we divide every number that is divisible by either of these factors or by their product; thus, we divide 10 by both 2 and 5; 12 by 2; 15 by 5; and 75 by 5. We next divide the second line in like manner by 2 and 3; and afterwards the third line by 5. By this process we collect the factors of the given numbers into groups; and the product of the factors at the left of the vertical is the least common multiple sought. 2 X 5 X2 X3 X 5300, Ans. 3. What is the least common multiple of 6, 15, 35, 42, and 70? Give explanation. 70; and whatever will contain 42 and 70 must contain 6 and 35. Hence we have only to find the least common multiple of the remaining numbers, 15, 42, and 70. From these examples we derive the following RULE. I. Write the numbers in a line, omitting any of the smaller numbers that are factors of the larger, and draw a vertical line at the left. II. Divide by any prime factor, or factors, that may be contained in one or more of the given numbers, and write the quotients and undivided numbers in a line underneath, omitting the 1's. III. In like manner divide the quotients and undivided numbers, and continue the process till all the factors of the given numbers have been transferred to the left of the vertical. Then multiply these factors together, and their product will be the least common multiple required. EXAMPLES FOR PRACTICE. 4. What is the least common multiple of 12, 15, 42, and 60? Ans. 420. 5. What is the least common multiple of 21, 35, and 42? Ans. 210. 6. What is the least common multiple of 25, 60, 100, and 125? Ans. 1500. 7. What is the least common multiple of 16, 40, 96, and 105 ? Ans. 3360. 8. What is the least common multiple of 4, 16, 20, 48, 60, and 72 ? Ans. 720. 9. What is the least common multiple of 84, 100, 224, and 300 ? Ans. 16800. Rule, first step? Second? Third? 10. What is the least common multiple of 270, 189, 297, 243? Ans. 187110. 11. What is the least common multiple of 1, 2, 3, 4, 5, 6, 7, 8,9? Ans. 2520. 12. What is the smallest sum of money for which I could purchase an exact number of books, at 5 dollars, or 3 dollars, or 4 dollars, or 6 dollars each? Ans. 60 dollars. 13. A farmer has 3 teams; the first can draw 12 barrels of flour, the second 15 barrels, and the third 18 barrels ; what is the smallest number of barrels that will make full loads for any of the teams? Ans. 180. 14. What is the smallest sum of money with which I can purchase cows at $30 each, oxen at $55 each, or horses at $105 each? Ans. $2310. 15. A can shear 41 sheep in a day, B 63, and C 54; what is the number of sheep in the smallest flock that would furnish exact days' labor for each of them shearing alone? fowl Ans. 15498. 16. A servant being ordered to lay out equal sums in the purchase of chickens, ducks, and turkeys, and to expend as little money as possible, agreed to forfeit 5 cents for every purchased more than was necessary to obey orders. In the market he found chickens at 12 cents, ducks at 30 cents, and turkeys at two prices, 75 cents and 90 cents, of which he imprudently took the cheaper; how much did he thereby forfeit? • Ans. 80 cents. CLASSIFICATION OF NUMBERS. Numbers may be classified as follows: 106. I. As Even and Odd. 107. II. As Prime and Composite. What is the first classification of numbers? What is an even number? An odd number? Second classification? A prime number? A composite number? 108. III. As Integral and Fractional. An Integral Number, or Integer, expresses whole things. Thus, 281; 78 boys; 1000 books. A Fractional Number, or Fraction, expresses equal parts of a thing. Thus, half a dollar; three-fourths of an hour; seven-eighths of a mile. 109. IV. As Abstract and Concrete. 110. V. As Simple and Compound. A Simple Number is either an abstract number, or a concrete number of but one denomination. Thus, 48, 926; 48 dollars, 926 miles. A Compound Number is a concrete number whose value is expressed in two or more different denominations. Thus, 32 dollars 15 cents; 15 days 4 hours 25 minutes; 7 miles 82 rods 9 feet 6 inches. 111. VI. As Like and Unlike. Like Numbers are numbers of the same unit value. If simple numbers, they must be all abstract, as 6, 62, 487; or all of one and the same denomination, as 5 apples, 62 apples, 487 apples; and, if compound numbers, they must be used to express the same kind of quantity, as time, distance, &c. Thus, 4 weeks 3 days 16 hours; 1 week 6 days 9 hours; 5 miles 40 rods; 2 miles 100 rods. Unlike Numbers are numbers of different unit values. Thus, 75, 140 dollars, and 28 miles; 4 hours 30 minutes, and 5 bushels 1 peck. What is the third classification? What is an integral number? A fractional number? What is the fourth classification? An abstract number? A concrete number? What is the fifth classification? A simple number? A compound number? Sixth classification? What are like numbers? Unlike numbers? FRACTIONS. DEFINITIONS, NOTATION, AND NUMERATION. 112. If a unit be divided into 2 equal parts, one of the parts is called one half. If a unit be divided into 3 equal parts, one of the parts is called one third, two of the parts two thirds. If a unit be divided into 4 equal parts, one of the parts is called one fourth, two of the parts two fourths, three of the parts three fourths. If a unit be divided into 5 equal parts, one of the parts is called one fifth, two of the parts two fifths, three of the parts three fifths, &c. The parts are expressed by figures; thus, Hence we see that the parts into which a unit is divided take their name, and their value, from the number of equal parts into which the unit is divided. Thus, if we divide an orange into 2 equal parts, the parts are called halves; if into 3 equal parts, thirds; if into 4 equal parts, fourths, &c.; and each third is less in value than each half, and each fourth less than each third; and the greater the number of parts, the less their value. When a unit is divided into any number of equal parts, one or more of these parts is a fractional part of the whole number, and is called a fraction. Hence 113. A Fraction is one or more of the equal parts of a unit. Define a fraction. |