Name the greatest common divisor— 6. Of 6 and 9. Of 12 and 10. PRINCIPLE. The product of the common factors of two or more numbers is the greatest common divisor of those numbers. WRITTEN PROBLEMS. Ex. Find the greatest common divisor of 15, 30 and 60. SOLUTION 1. 15=3×5 3x5 15 G. C. D. SOLUTION 2. 5)15, 30, 60 3)3, 6, 12 1, 2, 4 5x3=15 G. C. D. Explanation 1.-By examining the solution we find the only prime factors common to 15, 30 and 60 are 3 and 5; hence the greatest common divisor of these numbers is 3×5, or 15. Explanation 2.-5, being contained in all the numbers, is a common factor; 3, being contained in all the numbers, is also a common factor, and since the numbers have no other common factors, the greatest common divisor is 3× 5, or 15. From these solutions we derive the rule for finding the greatest common divisor of two or more numbers: RULE. Find the factors common to all the numbers, and take the product of these factors. Find the greatest common divisor 1. Of 15, 20, 30. 2. Of 16, 20, 24. Ans. 5. Ans. 4. Ans. 25. 10. A man has 2 logs which he wishes to cut into boards of equal length; one is 24 feet, and the other 16 feet long: how long can he cut the boards? Ans. 8 ft. 11. What is the greatest common divisor of $27, $36 and $72 ? Ans. $9. 12. What is the greatest equal lengths into which two trees can be cut, one being 105 feet in length and the other 84 feet? Ans. 21 feet. SECTION III. LEAST COMMON MULTIPLE. 65. A Multiple of a number is a number which is exactly divisible by that number. 66. A Common Multiple of two or more numbers is a number that is exactly divisible by each. Thus, 18 is a common multiple of 9 and 6, because it is divisible by each of them. 67. The Least Common Multiple of two or more numbers is the least number exactly divisible by each of them. Thus, 30 is the least common multiple of 10 and 15, because it is the least number exactly divisible by each of them. ORAL EXERCISES. What is a multiple— 1. Of 3? Of 4? Of 5? Of 10? Of 6? Of 20? PRINCIPLES.-1. Every multiple of a number contains the prime factors of that number. 2. The least common multiple of two or more numbers contains all the prime factors of each of the numbers, and no other factors. WRITTEN PROBLEMS. Ex. Find the least common multiple of 10, 15 and 20. SOLUTION. 10=2×5 15=3×5 20=2×2×5 L. C. M.=2×5×3×2=60 Explanation.—The least common multiple of the given numbers must contain the factors 2 and 5 to be divisible by 10; it must contain the factors 3 and 5 to be divisible by 15; it must contain the factors 2, 2 and 5 to be divisible by 20. Since the number 60 contains all these factors and no others, it is the least common multiple of 10, 15, 20. From the foregoing we derive the rule for finding the least common multiple of two or more numbers: RULE. Find the prime factors of the numbers, and take the product of these factors, using each the greatest number of times it occurs in any of the given numbers. What is the least common multiple 1. Of 15, 10 and 5? Ans. 30. Ans. 60. 10. Of 18 men, 16 men and 12 men? Ans. 144 men. 11. What is the smallest tract of land that may be cut into 6-acre, 5-acre or 4-acre lots? Ans. 60 acres. 12. What is the smallest sum for which I can hire workmen at 6, 8 or 9 dollars a week? Ans. $72. 13. What is the least common multiple of 10, 15, 20, 25 and 24? Ans. 600. CHAPTER IV. FRACTIONS. SECTION I. DEFINITIONS. If a unit be divided into two equal parts, each of these parts is called one-half; if into three equal parts, each part is called one-third; if into four equal parts, each part is called one-fourth; two of the four equal parts are called two-fourths, and so on. These parts of a unit-one-half, one-third, one-fourth, two-fourths, three-fourths, etc. are called fractions. 68. A Fraction is one or more of the equal parts of a unit. 69. A fraction is usually expressed by writing the numbers one above the other with a short horizontal line between them. Thus, three-fourths is written; two-thirds, , etc. 70. The number below the line is called the Denominator, and shows into how many equal parts the unit is divided. 71. The number above the line is called the Numerator, and shows how many of the equal parts are taken. Thus, in the denominator, 5, shows that the unit is divided into 5 equal parts; and 4, the numerator, shows that 4 of these parts are taken. 72. The numerator and the denominator are called the Terms of the fraction. EXERCISE. Read-,,, 3, 4, 4, 3, 4, 3, 1%. Write one-third, two-fourths, three-ninths, six-sevenths, five-sevenths, three-eighths, five-ninths, seven-tenths, thirteen-fifteenths, sixteen-fortieths. 73. A Common Fraction is one in which the numerator and the denominator are both expressed by figures. 74. Common fractions are either proper or improper. 75. A Proper Fraction is one whose numerator is less than its denominator; as,,, etc. 76. An Improper Fraction is one whose numerator is equal to or greater than its denominator; as,,,, etc. 77. A Mixed Number is one consisting of a whole number and a fraction. |