Another Method of Finding the Least Common Multiple. The preceding Rule is given to show the composition of the least common multiple. The following is the common Rule; and is preferable in a practical point of view. RULE X V. $86. To find the least common multiple of two or more numbers. 1. Set the numbers in a line, from left to right, and divide any two or more of them by any prime number, greater than a unit, that will divide them without a remainder, placing the quotient and the undivided numbers in a line below. 2. Divide any two or more of the numbers in the lower line, as before; and so on, until no two numbers in the lowest line, can be so divided. The product of the divisors and numbers in the lowest line, will be the least common multiple of the given numbers. 3. If no two of the given numbers can be divided as above, the product of all the given numbers will be their least common multiple. EXAMPLE. To find the least common multiple of 6, 12, and 15. We divide 6 and 12 by 2; and place 15 in a line with the quotients, since 15 will not divide by two without a remainder. We then divide 3, 6, and 15, by 3; when we find that no two of the numbers, 1, 2, 5, in the lowest line, can be divided by any number greater than a unit, without a remainder. Taking now the divisors, 2 and 3, and the numbers 2 and 5 in the lowest line, omitting the 1, as not affecting the product, we have 2×3×2×560, for the least common multiple of 6, 12, and 15. This method is the same in principle with that of Rule XIV. The divisors and numbers in the lowest line, are necessarily prime factors of the given numbers; and are the smallest selection of such factors that includes the factors of each given number. D EXERCISES. 1. Find the least common multiple of 4, 2. Find the least common multiple of 9, 3. Find the least common multiple of 21, 4. Find the least common multiple of 6, 5. Find the least common multiple of 8, 6. Find the least common multiple of 15, 7. Find the least common multiple of 24, 8. Find the least common multiple of 5, 9. Find the least common multiple of 6, 10. Find the least common multiple of 11, 6, 8, and 10. Ans. 252. Ans. 1560. 7, 2, and 17. Ans. 714. 4, 5, and 19. Ans. 4180. APPLICATION OF COMMON MULTIPLE. 1. What is the smallest sum of money for which a person could purchase, either a number of mules at 32 dollars a head, or a number of cows at 14 dollars a head,—the same sum to be employed in either purchase? It is evident that the number of dollars to be employed, is the least common multiple of 32 and 14. Ans. 224 dollars. 2. A can build 7 rods of fence in a day; B can build 9 rods; and C 12 rods, in a day. What amount of fencing would afford a number of whole days' work for any one of the three? Ans. 252 rods, or 504 rods, or 756 rods, &c. 3. If one team can haul to market 10 barrels of flour; another, 12 barrels; and another, 15 barrels;-what number of barrels would make a number of full loads for any of the three teams? Ans. 60 barrels, or 120 barrels, or 360 barrels, &c. 4. How many bushels of wheat would fill a number of barrels, each containing 3 bushels; or a number of sacks, each containing 4 bushels; or a number of hogsheads, each containing 15 bushels;-the quantity to be the same in each case? Ans. 60 bushels, or 120 bushels, or 180 bushels, &c. 5. What is the smallest sum for which I could purchase a number of mules, at 35 dollars a head; or a number of horses, at 45 dollars a head; and what number of each could I purchase for that sum? Ans. 315 dollars; 9 mules, or 7 horses. EXERCISES ON CHAPTER III. Find the value of each of the following expressions — multiplying and dividing by means of factors. 1. (1000+250-30+375)X(81-9). 7. Resolve 780 into its prime factors. Ans. 114840. Ans. 314784. 59 49 Ans. 25133 45 Ans. 93121 Ans. 2, 2, and 109. Ans. 2, 2, 5, 3, and 13. Ans. 2, 2, 3, 3, 3, 3, and 3. 8. Resolve 972 into its prime factors. 9. Resolve 1275 into its prime factors. Ans. 3, 5, 5, and 17. 10. Resolve 2000 into its prime factors. Ans. 2, 2, 2, 2, 5, 5, and 5. 11. Find the greatest common measure of 124, 200, and 350. Ans. 2. 12. Find the greatest common measure of 325, 240, and 460. Ans. 5. 13. Find the greatest common measure of 270, 800, and 960. Ans. 10. 14. Find the least common multiple of 14, 25, 8, and 20. Ans. 1400. 15. Find the least common multiple of 8, 36, 9, and 17. Ans. 1224. 16. Find the common multiples of 15, 20, 32, and 75. Ans. 2400, 4800, 7200, &c. 17. Find the common measures of 300, 400, 500, and 600. Ans. 2, 4, 5, 10, 20, 25, 50, and 100. 18. Find the common multiples of 24, 36, 60, and 84. Ans. 2520, 5040, 7560, &c. 19. Find the greatest common measure, and also the least common multiple, of 74 and 126, Ans. Greatest com. meas. 2; least com. mult. 4662. 20. The junior class in a school consists of 132 students, and the senior of 99. How might each class be divided, so that the whole school should be disposed in equal sections? Ans. Into sections of 3, 11, or 33. 21. For what sum of money could a carpenter hire journeymen for one month, at 15 dollars, 21 dollars, or 24 dollars each, allowing the whole sum to be thus expended? Ans. 840 dollars, or 1680 dollars, &c. 22. What is the smallest sum of money for which I could purchase a number of plows at 14 dollars each, or a number of carts at 30 dollars each, or a number of wagons at 90 dollars each? Ans. 630 dollars. Ans. 37 gallons. 23. A wine merchant has 111 gallons of Madeira, 185 gallons of Port, and 259 gallons of Malaga, with which he wishes to fill a number of casks, all containing the same number of gallons, and without mixing the different kinds of wine. What must be the contents of each cask? 24. A has 413 dollars, B 531 dollars, and C 590 dollars; and they agree to purchase horses, at the same price per head, provided each man can thus invest all his money. How many horses could each man purchase? Ans. A could purchase 7, B 9, and C 10. 25. An island is 200 miles in circumference, and three persons, A, B, and C, start together, and travel the same way \around it. A goes 20 miles per day, B 25, and C 40 miles per day. In what time would they all come together again at the same point from which they started? First find the number of days it would require each person to go around the island. Ans. 40 days. CHAPTER IV. PRELIMINARY DEFINITIONS AND PRINCIPLES -REDUCTION OF FRACTIONS. FRACTIONS. $ 87. A Fraction is an expression of one or more of the equal parts into which any quantity may be divided. One half is one of the two equal parts,-One third is one of the three equal parts,-Two thirds are two of the three equal parts, and so on, of any quantity. What is meant by one fourth of a quantity? By three fourths of a quantity? By one fifth of a quantity? By four fifths? Any quantity consists of how many halves of that quantity? Of how many thirds? Of how many fourths? Of how many tenths? Which is the greater part, one half or one third of a quantity? fourth or one seventh? One ninth or one fifth? hundredth? One sixth or one sixtieth? How many is one half of 2? One third of 3? Three fourths of 4? One One tenth or one Two thirds of 3? One fifth of 5? Five sixths of 6? One seventh of 7 Three eighths of 8? Four ninths of 9? If the 2 halves of any quantity were each divided into 2 equal parts, into how many equal parts would the whole quantity be divided? What would each of those parts be called? One half is how many fourths? If the 3 thirds of any quantity were each divided into 2 equal parts, into how many equal parts would the whole quantity be divided? What would each of those parts be called? One third is how many sixths? If the 4 fourths of any quantity were each divided into 3 equal parts, into how many equal parts would the whole quantity be divided? What would each of those parts be called? One fourth is how many twelfths? Numerator and Denominator. § 88. A fraction is denoted by two numbers, one above the other, with a line between them. The upper number is called the numerator; and the lower, the denominator. The numerator shows the number of cqual parts in the fraction; the denominator shows the number of such parts in the whole quantity divided. Thus, three fourths; 3 is the numerator, and 4 the denominator. |
