5. Divide 16×64× 81 by 72×3×8×6. 10. A farmer hired 8 men for 6 months at $18 a month; he paid them with hay at $12 a ton: how many tons did they receive? 11. A merchant bought 12 casks of rice, each containing 95 lb. at 10 cents a pound, and paid for it with 3 barrels of butter at 19 cents a pound: how many pounds in a barrel? GREATEST COMMON DIVISOR. 86. A Common Divisor of two or more numbers is any number that will divide each without a remainder. The Greatest Common Divisor of two or more numbers is the greatest number that will divide each without a remainder. NOTE.-Numbers may have several common divisors, but only one greatest common divisor. EXAMPLE.-What is the greatest common divisor of 42 and 54? SOLUTION.-Resolve each number into OPERATION. 42=2×3×7 54-2x3x3x3 2x3=6 prime factors. The prime factors of 42 are 2, 3, and 7; the prime factors of 54 are 2, 3, 3, and 3. The factors 2 and 3 are the only ones common to both numbers; hence, their product, 6, is the greatest number that will divide each of them. CONCLUSION.-Therefore, the greatest common divisor of 42 and 54 is 6. Hence, the Rule.-Resolve the given numbers into their prime factors. Multiply the common factors, and their product will be the greatest common divisor. EXAMPLES FOR PRACTICE. Find the greatest common divisor 1. Of 24 and 36. 7. Of 312 and 429. 2. Of 72 and 99. 3. Of 96 and 132. 4. Of 221 and 323. 5. Of 576 and 276. 6. Of 396 and 480. 8. Of 378 and 490. 9. Of 630 and 585. 10. Of 1360 and 1408. 11. Of 48, 63, and 111. 12. Of 96, 144, and 129. LEAST COMMON MULTIPLE. 87. A Common Multiple of two or more numbers is a number that can be divided by each without a remainder. Thus, 90 is a common multiple of 9 and 15. The Least Common Multiple of two or more numbers is the least number that can be divided by each without a remainder. Thus, 15 is the least common multiple of 3 and 5. A common multiple must contain the prime factors of each number.. Thus, 90 contains 3×3, the factors of 9; and 3×5, the factors of 15. The least common multiple must contain each factor only so many times as it occurs in any one number. Thus, 90 is not the least common multiple of 9 and 15, because it contains the factor 2, which is not found in 9 or 15. EXAMPLE.-What is the least common multiple of CONCLUSION.-Therefore, the least common multiple of 8, 12, and 15 is 120. Hence, the Rule.-Resolve the numbers into their prime factors. The product of each factor, taken as many times as it oceurs in any one number, will be the least common multiple. EXAMPLES FOR PRACTICE. Find the least common multiple 1. Of 12 and 20. 2. Of 15 and 35. 3. Of 52 and 63. 4. Of 70 and 84. 6. Of 21, 80, and 120. 7. Of 32, 48, and 80. 8. Of 54, 121, and 66. 9. Of 76, 85, and 216. 10. Of 46, 69, and 253. QUESTIONS FOR REVIEW. What is an integer? How are integers divided? What is a prime number? A composite number? When are numbers prime to each other? Is 3 a prime or a composite number? 9? 13? 15? 26? 31? Are 5 and 8 prime to each other? 6 and 9? 15 and 21? What is a factor? What is a prime factor? A composite factor? Mention a prime factor of 24. A composite factor. Is 6 a prime or a composite factor of 30? What kind of a factor is 5? What is a divisor? What is said of the terms factor, divisor, and measure? By what is any composite number divisible? What is a multiple? Of what numbers is 12 a multiple? Give several multiples of 12. Of 6. What are the prime factors of any number? Give the rule for finding them. What numbers are divisible by 2? By 4? By 5? By 3 or 9? Define cancellation. Upon what does it depend? The rule. What is a common divisor? The greatest common divisor? Can numbers have more than one common divisor? Can they have more than one greatest common divisor? Give two common divisors of 24 and 48. Give the greatest common divisor. Give the rule for finding the greatest common divisor. What is a common multiple? The least common multiple? Give examples of each. Give the rule for finding the least common multiple. What is reduction? How many kinds of reduction? What is the rule for each? For what is dry measure used? Give the table. For what is long measure used? Give the table. many feet make a fathom? How How many miles make a league? How many inches a hand? How long is a surveyor's chain? How is it divided? For what is wine measure used? Ale or beer measure? Give the table for wine measure. How is cloth measured? Give the table for solid measure. What is a cube? A square? How find the area of a square? What is a rectangle? What is a cord of wood? Give the time table. What is the length of a year? How many days has January? February? How many days are considered a month? Name the four seasons. Give the months belonging to each. How many sheets of paper in a quire? How many pounds of flour in a barrel? Pounds of corn in a bushel? Of oats? Of wheat? What year will be the next leap year? What is Federal Money? Give the denominations. The table. Which denominations are most used? What is a coin? Which Of nickel? coins are made of gold? Of silver? How are dollars and eagles read? How dimes and cents? What place do mills occupy? How is a number multiplied by 10? By 100? 1000? How, divided?. FRACTIONS. 88. A Fraction is an expression denoting one or more equal parts of a unit. When a unit is divided into two equal parts, one of these parts is called one-half (1). When a unit is divided into three' equal parts, one of these parts is called one-third (); two parts, twothirds (). When a unit is divided into four equal parts, one of these parts is called one-fourth (4); two parts, twofourths (); three parts, three-fourths (4). The Denominator of a fraction is that number which shows into how many parts a unit is divided. The Numerator is the number that shows how many of these parts are taken. In one-half (1), 2 is the denominator, and 1 is the numerator; in three-fourths (), 4 is the denominator, and 3 is the numerator, etc. The numerator and denominator are called the terms of the fraction. 89. There are two kinds of fractions, common and decimal. A Common Fraction is one whose denominator may be any number. Common fractions are expressed by two numbers, one above the other, with a horizontal line between. TO READ COMMON FRACTIONS:-Read the number of parts, as shown by the numerator; then the name of the part, as shown by the denominator. |
