2. What number is a multiple of 18? 24? 35? 45? 44? 60? 100? 250? NOTE. The teacher should show that a number has any number of multiples. 3. What number is a common multiple of 3 and 4? 4 and 5? 6 and 8? 5 and 6? 4. What number is a common multiple of 7 and 5? 3 and 9? 3, 4, and 6? 4, 8, and 12? NOTE.-The teacher should show that two or more numbers have any number of common multiples. 5. What is the least common multiple of 3 and 4? 5 and 6? 3, 6, and 12? 2, 4, and 8? 6. What is the least common multiple of 3, 5, and 10? 2, 5, and 10? 2, 3, 5, and 10? WRITTEN EXERCISES. 1. What is the least common multiple of 12, 18, and 30? Resolve the numbers into their prime factors, and select all the different factors, repeating each as many times as it is found in any numthe factor 3, twice in product of 2 × 2 × 3 ber. The factor 2 is found twice in 12; What is the least common multiple of 2. 12, 15, and 20? 7. 24, 72, 18, 48? 9. 75, 150, 300? Art. 53. A Multiple of a number is any number which it will exactly divide. NOTE.-Every number is an exact divisor of its product by an integer. is A Common Multiple of two or more numbers any number which each of them will exactly divide. The Least Common Multiple of two or more numbers is the least number which each of them will exactly divide. Art. 54. RULE.-To find the least common multiple of two or more numbers, Resolve the numbers into their prime factors, and then select all the different factors, taking each the highest number of times it is found in any number. Multiply the factors thus selected; their product will be the least common multiple. QUESTIONS FOR REVIEW. What is meant by the divisor of a number? When is a divisor a common divisor? Define a common divisor. What is the greatest common divisor of two or more numbers? How is it found? By what may every number be divided? What is a prime number? A composite number? What is an even number? An odd number? What is meant by the factor of a number? A prime factor? A composite factor? How may a composite number be resolved into prime factors? What is a multiple of a number? When is a multiple a common multiple? Define a common multiple. What is the least common multiple of two or more numbers? Give the rule for finding the least common multiple. What is the difference between a divisor and a multiple? The Idea of a Fraction developed. 1. If a melon be cut into two equal pieces, what part of the melon will one piece be? 2. How many halves in a melon? How many halves in any thing? 3. If a melon be cut into four equal part of the melon will one piece be? Three pieces? 4. How many fourths in an apple? fourths in any thing? pieces, what Two pieces? How many 5. Which is the greater, one half or one fourth of an apple? How many fourths equal one half? 6. If a cake be cut into three equal pieces, what part of the cake will one piece be? 7. How many thirds in a cake? How many thirds in any thing? 8. If a cake be cut into six equal pieces, what part of the cake will one piece be? Two pieces? Three pieces? Four pieces? Five pieces? 9. How many sixths in any thing? 10. Which is the greater, one third or one sixth of a cake? How many sixths equal one third? 11. A single thing is a unit. How many halves in a unit? How many thirds? How many fourths? How many sixths? 12. What is meant by one third? Ans. One third is one of the three equal parts of a unit. 13. What is meant by two thirds? One fourth? Three fourths? One sixth? Three sixths? 14. Which is the greater, two thirds or a unit? Five thirds or a unit? Three fourths or a unit? Four fourths or a unit? Art. 55. Such parts of a unit as two thirds, three fourths, five sixths, etc., are called Fractions. A fraction may be expressed by two numbers, one written under the other, with a horizontal line between them; as, ., . The number below the line denotes the number of equal parts into which the unit is divided. It is called the Denominator. The number above the line denotes the number of equal parts taken. It is called the Numerator. Read the following fractions. How is the unit di vided, and how many parts are taken in each case? Art. 56. A Fraction is one or more of the equal parts of a unit. Art. 57. A fraction is expressed by two numbers, called the Numerator and the Denominator. The Denominator of a fraction denotes the number of equal parts into which the unit is divided. The Numerator of a fraction denotes the number of equal parts taken. The numerator and denominator are called the Terms of a fraction. LESSON II. Integers and Mixed Numbers reduced to Fractions. 1. How many thirds in an apple? How many thirds in 2 apples? 2. How many fourths in 3 pears? SOLUTION.-In 1 pear there are 4 fourths, and in 3 pears there are three times 4 fourths, which is 12 fourths. There are 12 fourths in 3 pears? 3. How many sixths in 3 oranges? In 5 oranges? 6 oranges? 8 oranges? |