given on page 47. Find the g. c. d. of the following by the method Prove the work by the method. g. Find the sum of the six answers, a to f. Find the g. c. d. of 120, 180, and 250 by the method given on page 47. Prove the work by first finding the g. c. d. of two of the numbers, and then of the g.c.d. found and the third number, employing the method given on page 48. 7. Find the sum of the four answers, h to k. Remembering that the g. c. d. of any two numbers cannot exceed their difference, find by inspection and trial the g. c. d. of the following: m. 24, 82, 26, 96 o. 11, 15, 48, 99 n. 45, 55, 9265 p. 275, 625, 300 Find the sum of the four answers, m to p. HALL'S PRACT. ARITH. 4 To find the least common multiple (1. c. m.) of two or more numbers: 1. Find the 1. c. m of 24, 35, 36, and 50. (2 × 2 × 2)(3 × 3)(5 × 5) × 7 = 12600 RULE. Resolve each number into its prime factors. Take as factors of the 1. c. m. the greatest number of 2's, 3'8, 5'8, 7's, etc., found in any one of the numbers. Find the 1. c. m. of the following: g. Find the sum of the six answers, a to f. Observe that the 1. c. m. must contain every prime factor common to two or more of the numbers and every uncommon prime factor found in any of them. Hence the following: 2. Find the 1. c. m. of 21, 30, 35. OPERATION 321 30 35 EXPLANATION The prime factors common to two or more 35 of the numbers are 3, 7, and 5. The uncommon prime factor is 2, found in the number 30. The 1. c. m. is the product of 3, 7, 5, and 2, which is 210. 5 1 1. Find several common multiples of 6 and 8. b. Find the 1. c. m. of 18 and 30. 3. Find several common divisors of 12 and 18. c. Find several common divisors of 48 and 60. 4. Find the g. c. d. of 12 and 18. d. Find the g. c. d. of 48 and 60. 5. Name two numbers whose product is their 1. c. m. 6. Find the 1. c. m. of the following: m.* e. 10, 20, 24 f. 20, 30, 40, 50 g. 50, 60, 70 i. 18, 12, 39 k. 21, 45, 63 h. 27, 54, 63, 72 j. 72, 96, 144, 120 7. 75, 125, 150, 175 Remembering that the g. c. d. of two numbers is never greater than their difference, find by inspection and trial the g. c. d. of the following: 7. Remembering that 25 is contained exactly 4 times in 100, tell without the aid of a pencil how many times 25 is contained in 825. 8. Remembering that 163 is contained exactly 6 times in 100, tell without the aid of a pencil how many times 16 is contained in 933. * The numbers must contain no common factor; that is, they must be prime to each other. 1. 643,265,245,350. Without dividing, tell whether this number is exactly divisible by 9; by 5; by 10; by 25; by 50; by 121; by 163.* A number is made up of the following prime factors: 2, 2, 3, 3, 5, 7. a. What is the number"? b. Is the number whose prime factors are given above exactly divisible by 35? c. How many times is 70 (2 × 5 × 7) contained in the number? d. How many times is 45 (3 × 3 × 5) contained in the number? e. How many times is 36 (2 × 2 × 3 × 3) contained in the number? Resolve the number 836 into its prime factors, and write these factors on paper or on the blackboard. Then answer the following: f. How many times is 19 contained in 836? g. How many times is 209 (11 x 19) contained in 836? h. How many times is 418 (19 x 11 x 2) contained in 836? i. Multiply 8246 by 39. plying 8246 by 13, and the Prove the work by multiproduct thus obtained by 3. j. Multiply 9463 by 48. Prove the work by multiplying 9463 by 12, and the product thus obtained by 4. k. Multiply 3462 by 96. Prove the work by multiplying 3462 by 8 and the product thus obtained by 12. * A careful study of pages 43-47 will enable the pupil to make the statements called for with little hesitation. FRACTIONS There are three aspects in which fractions should now. be considered: I. THE FRACTIONAL UNIT ASPECT The numerator tells the number of things, and the denominator indicates their name. In the fraction there are 5 things (magnitudes) called sevenths. In this fraction there are 5 fractional units, each of which is 1 seventh of another unit called the unit of the fraction. The denominator shows the number of parts into which the unit of the fraction is divided; the numerator shows how many of these parts (fractional units) are taken. II. THE DIVISION ASPECT The numerator is thought of as a dividend, the denominator, a divisor, and the fraction itself, a quotient: thus, in the fraction , the dividend is 5, the divisor, 7, and the quotient, . In the case of an improper fraction, as, it may be more readily seen by the pupil that the numerator is the dividend, the denominator, the divisor, and the fraction (2) the quotient; but the division relation is in every fraction, whether proper or improper, common or decimal, simple or complex. III. THE RATIO ASPECT The numerator of a fraction may be thought of as an antecedent, the denominator, a consequent, and the fraction itself, a ratio: thus, in the fraction 4, 5 is the antecedent, 7 the consequent, and the ratio. This relation may be more readily seen by the pupil in the case of an improper fraction. In the fraction 2, 12 is the antecedent; 4 the consequent; 12, or 3, the ratio. |