Numbers are prime to each other when they have no common divisor, as 7 and 12. 11. Write five pairs of numbers that are prime to each other. LESSON 25 1 11111111111111111111111111111 Thirty32 82 82 82 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 82 32 32 32 32 32 32 seconds 1 1. = how many fourths ? How many eighths ? How many sixteenths? How many thirty-seconds ? 2. By what number must you multiply both numerator and denominator of } to make it equal ? $? 16? 19? 3. Does the diagram show you that} = = 4 = 16 = 19? = how many sixteenths ? By what number must you multiply both terms of to make it equal 16? 5. 4 = how many sixteenths ? What must you do to both numerator and denominator of to make it equal ? 6. If you multiply both numerator and denominator of Å by 8, do you change the value of the fraction ? 7. Does the diagram show that f= 66 ? =16? 1=31? 8. Does multiplying both numerator and denominator of a fraction by the same number alter the value of the fraction? 9. Change to 8ths: 1, 3, 4, 4, 11, 14, 2, 27. 10. Change to 16ths: 1, 1, 4, 11, 14, 14, 5, f. 11. Change to 32ds : , , 16, 5, 1, 16 , 13. 12. Change to 12ths : 1, , , 1, , , , 1. 13. Change to 24ths : 1, , , 1, 4, 6, 5, Tz. 14. } equals how many 6ths ? 9ths ? 12ths ? 15ths ? 15. Compare of a foot with 12 of a foot. 16. Compare 1 of a dozen with } of a dozen. LESSON 26 1. By what number must you divide both numerator and denominator of 6 to make it equal ? 4 ? 2. By what number must you divide both numerator and denominator of 12 to make it equal & ? 3. Does the diagram on p. 31 show you that 12 = f= ? 4. What must you do to both numerator and denominator of 4 to make it equal ? 5. If you divide both terms of 18 by 2, what fraction will you get ? Have you changed the value of the fraction? 6. Does dividing both numerator and denominator of a fraction by the same number alter the value of the fraction? 7. Change to 4ths ; to 6ths ; 12 to 4ths; * to 4ths; to 4ths ; 12 to 8ths ; 19 to 6ths ; 12 to 16ths. 8. Compare ia of a day with ģ of a day. 9. What is the G. C. D. of 25 and 40 ? 10. Divide both numerator and denominator of 46 by their G. C.D. What is your answer? 11. Divide the numerator and denominator of by 6. What fraction do you get for your answer? 12. Is 6 the G. C. D. of 36 and 42 ? Why? 13. By what number must you divide both terms of 18 to make it equal 1 ? Are the numerator and denominator of 18 prime to each other? Why? Are the numerator and denominator of 4 prime to each other? Why? When are numbers prime to each other? (Lesson 24.) When the numerator and denominator of a fraction are prime to each other, the fraction is said to be in its lowest terms. Change the following fractions to their lowest terms : 14. %, 4, 12, 15, 211, 21 15. , 12, 1, 14, 18, d. LESSON 27 2. 12 21 15. 48 Divide the numerator and denominator of each of the following fractions by their G. C. D.: 25 : 5= 5 1. 5. 34 9.34 13. 6. 21 14. 48 7. } 11. 88 4. 21 8. 15 12.38 16. 11 NOTE. — This process is called changing fractions to lowest terms. 17. Write five equivalents for 1. Thus, 1 = 4, , etc. 18. Write five equivalents for }, j, 1, , . 19. Change to 4ths : 8, 12, 16, 18, , 620. Change to 9ths: Ķ, 8, 13, 19, 37, 25, 34. 21. Change to 10ths: 1, , , 28, 28, 29, 13. GRAD. ARITH. v.-3 22. Write all the divisors of 36, 54, 63, 84. LEAST COMMON MULTIPLE LESSON 28 When a number can be divided by another without a remainder, it is called a Multiple of that number. Thus, 9 is a multiple of 3. 1. Name the smallest number that is a multiple of 3. Name all the multiples of 3 to 27; 4 to 32 ; 5 to 35; 6 to 48. 2. Name a multiple of 2 and 3; that is, a number that is divisible by 2 and 3. 3. Name a multiple of 3 and 4; 5 and 6; 6 and 8. 4. Name the least number that can be exactly divided by 4 and 6; 3 and 6; 6 and 15; 3 and 8. The smallest number that can be divided by two or more numbers without a remainder is called the Least Common Multiple (L. C. M.) of those numbers. Thus, 18 is the least common multiple of 6 and 9, since it is the least number that can be exactly divided by each of them. Name at sight L. C. M. of: 5. 8 and 12 ; 6 and 9; 10 and 15 ; 12 and 16; 2, 3, 4, and 6; 16 and 24 ; 12 and 15; 3, 6, 9, and 12; 2, 7, and 14 ; 12, 4, and 18. The L. C. M. of numbers prime to each other is simply their product. Thus, the L. C. M. of 3 and 5 is 15 ; of 2, 3, and 7 is 42. 6. Name the L. C. M. of 4 and 5; 3 and 8; 4 and 5 7 and 8; 6 and 11. 7. Name all the prime numbers to 100. 8. What is the G. C. D. of 19 and 38 ? 13 and 26 ? 9. Name the multiples of 8 to 96 ; 9 to 108. 10. Name two equal factors of 9; 16; 36; 49; 64; 100. 11. Name the L. C. M. of 3, 10, and 12; 3, 12, 30, and 5. LESSON 29 The L. C. M. of two or more numbers must be composed only of the prime factors of each of the numbers. 1. Find the L. C. M. of 6, 9, and 24. MODEL. 6 2 x 3 9= 3 x 3 24 2 x 2 x 2 x 3 The L. C.M. must contain all the prime factors of 24 (2 x 2x2x3). It must contain the factors of 9 (3 x 3). We already have one 3 in the factors of 24. Hence we retain one 3 and reject the other. The L. C. M. must also contain the factors of 6 :(2 x 3). As we have these factors already in 24, we reject them also. The factors and the numbers they represent may be shown as follows: 24 L.C.M., 72 = 2 * 2 * 2 * 3 * 3. 6. 9 |