10. 132 and 144 ? 11. 49 and 72? 12. 35 and 707 13. 78 and 91 ? 14. 36 and 72? 15. 40 and 75 ? 16. 84 and 97 ? 17. 63 and 98? 18. 12 and 25 ? 19. 17 and 69 ? 20. 150 and 210? 21. 143 and 176? 22. 462 and 693 ? 72. Method for Large Numbers. (a.) The greatest common divisor of two numbers is the same as the greatest common divisor of the smaller number, and the remainder left after dividing the larger number by the smaller.. ILLUSTRATIONS. — The greatest common divisor of 36 and 102 is the same as the greatest common divisor of 36, and the remainder, 30, which is left after dividing 102 by 36. Again : the greatest common divisor of this remainder, 30, and the smaller number, 36, is the same as the greatest common divisor of 30 and the remainder, 6, which is left after dividing 36 by 30. Now, as 6 divides 30, it is the greatest common divisor of 30 and 36, and also of the two original numbers. 1. What is the greatest common divisor of 3473 and 8909 ? 4473)8909(2 6946 1963)3473(1 1963 1510 = 2 X 5 X 151. 151)1963(13 151 Solution. As the factors of these numbers are not readily recognized, we divide the greater by the less, which gives 2 for a quotient and 1963 for a remainder. We have now to find the greatest common divisor of 1963 and 3473. As the factors of 1963 are not readily recognized, we divide 3473 by it, which gives 1 for a quotient and 1510 for a remainder. Hence, we have now 453 453 to find the greatest common divisor of 1510 and 1963. But 1510 = 2 x 5 X 151, of which factors only 151 is a factor of 1963. Hence, 151 is the greatest common divisor required. (b.) What is the greatest common divisor of — 2. 221 and 507? 7. 833 and 1547 ? 73. To find the Least Common Multiple. (a.) A MULTIPLE of a number is any number which contains it as a factor. (b.) A Common MULTIPLE of two or more numbers is any num-. ber which contains them all as factors. (c.) The LEAST COMMON MULTIPLE of two or more numbers is the least number which contains them all as factors. (d.) From these definitions, and the principles previously established, it follows that 1st. A multiple of a number must contain all the prime factors of that number. 2d. A common multiple of two or more numbers must contain all the prime factors of those numbers. 3d. The least common multiple of two or more numbers must be the least number that contains all the prime factors of those numbers. (e.) To find the least common multiple of two or more numbers, we multiply one of them by such prime factors of the other numbers as it does not contain. 1. What is the least common multiple of 270 and 504? SOLUTION. The least common multiple of 270 and 504 is the least num. ber which contains all the prime factors of 270 and 504. 270 2 X 38 X 5. We must have all the prime factors of 270, or, which is the same thing, 270 itself. We must also have all the prime factors of 504, which are 23 X 39 X 7. But as in 270 we have already taken the 3's and one of the 2's, we shall only have to introduce the two remaining 2's and the 7, i. e. 22 or 4, and 7. Hence, the least common multiple of 270 and 504 is 270 X 4 X 7 : 7560. NOTE. Had we first taken the factors of 504, or 504 itself, the work would have taken the form, 504 X 3 X 5 = 7560. What is the least common multiple of 2. 12 and 18 ? 12. 16 and 27 ? 22. What is the least common multiple of 21, 24, 28, and 35 ? SOLUTION. - The least common multiple of 21, 24, 28, and 35, is the smallest number which contains all the prime factors of these numbers. 21 = 3 X 7. 28 = 22 X 7. 24 = 3 X 2'. 35 = 5 x 7. Beginning with 35, we must have all its prime factors, or, which is the same thing, 35 itself. We must also have all the prime factors of 28, which are 22 X 7. But as we have already taken 7, we have only to introduce the 22 or 4, giving 4 X 35. We must also have the prime factors of 24, which are 23 X 3. As we have already taken 22, we have only to introduce 2 X 3, or 6, giving 6 X 4 X 35. We must also have the prime factors of 21, which are 3 X 7. But as in taking 35 we took the factor 7, and in taking 6 we took the factor 3, we have no new factor to introduce. Hence, the least common multiple required is 6 X 4 X 35, or 840. NOTE. — We might have begun with any of the other numbers instead of 35, but it is usually most convenient to begin with the largest number. The careful student will doubtless discover abbreviated methods of finding the least common multiple. He may adopt any method by which he can find the required factors. (f.) What is the least common multiple of 23. 4, 6, 8, and 10? 33. 18, 24, 30, and 36 ? 24. 3, 5, 9, and 10? 34. 9, 11, 14, and 16 ? 25. 5, 7, 9, and 12? 35. 3, 5, 7, 11, and 13? 26. 2, 4, 6, 9, and 18? 36. 28, 42, and 70 ? 27. 8, 12, 15, and 20 ? 37. 25, 50, 75, 125, and 150? 28. 2, 3, 4, 5, and 6? 38. 42, 49, and 56 ? 29. 5, 6, 7, 8, and 9? 39. 4, 8, 16, 32, and 64? 30. 6, 7, 8, 9, and 10? 40. 54, 81, 108, and 135? 31. 7, 8, 9, 10, and 11? 41. 39, 51, 52, and 68? 32. 8, 9, 10, 11, and 12? 42. 60, 76, 114, and 120 ? SECTION X V. FRACTIONS. 74. Definitions and Explanations. (a.) Such parts as are obtained by dividing any unit whatever into equal parts are called Fractional Parts, and the numbers expressing them are called FRACTIONS. (b.) Halves are such fractional parts as are obtained by dividing a unit into two equal parts. (c.) THIRDS are such fractional parts as are obtained by dividing a unit into three equal parts. (d.) In like manner we may define FOURTHS, FIFTAS, SIXTHS, SEVENTHS, etc. (e.) The following definitions are sometimes more convenient in their application than the preceding. (f.) Halves are equal parts of such kind that two of them are equal to a unit. (g.) In like manner we may define thirds, fourths, fifths, etc. (h.) Fractional parts are generally expressed by drawing a line above the number which shows how many of them must be taken to equal a unit. ILLUSTRATIONS.—We may express fourths by writing the figure 4 with a line above it, thus, 4. In like manner, o means ninths, 17= twelfths, etc. (j.) The number which thus shows the kind or denomination of fractional parts is called a DENOMINATOR. (k.) To show how many fractional parts are taken or considered, a figure called the NUMERATOR is written above the denominator. ILLUSTRATION. — To express 3 sevenths, we write the numerator 3 above the denominator 7, thus, – (1.) Write each of the following fractions in figures : 1. Three-fifths. 4. Six-seventeenths. 2. Five-ninths. 5. Twenty forty-firsts. 3. Twelve-fifths. 6. Fifty-six eighths. 7. What does the fraction express ? ANSWER. - Three-fifths expresses the value of three such parts as are obtained by dividing a unit into five equal parts. (m.) In the same manner explain each of the following: 8. . 14. .36 13. .07 MODEL.— Three-fifths expresses the value of three equal parts such that five of them would equal a unit. (0.) A Mixed NUMBER is a whole number and a fraction. ILLUSTRATION. — 5, which is read “five and two-thirds," is a mixed number. 10. io 12. 13 15. 5 75. Recapitulation. (a.) A Fraction is a number which expresses one or more fractional parts. (b.) FRACTIONAL PARTs are such parts as are obtained by dividing a unit into equal parts, or — |