85 b. To add United States Currency. 1. Add $25, $35, $45. Suggestion. - Add as in whole numbers. 2. Find the sum of $271, $127, $712. Suggestion. — Subtract as in whole numbers. 5. From 73 cents subtract 37 cents. 6. From $73.37 subtract $37.73. 7. From $91.91 subtract $19.19. 85 d. To multiply United States Currency. Suggestion. - Multiply as in whole numbers. Suggestion. -Divide as in whole numbers. 14. Divide $91.15 by 5. 15. Divide $144.24 by 4. 16. Divide $81.27 by 9. 17. Divide $169.26 by 13. 18. Divide $216.56 by 8. 19. Divide $288.48 by 12. 20. Divide $576.96 by 24. 21. Divide $512.32 by 16. 22. Divide $2142.84 by 21. 23. Divide $4284.84 by 42. 24. What does 1 man earn, if 12 men earn $172.80? 25. If 12 yards cost $42.60, what did 1 yard cost? 26. Find the cost of 1 ox, if 17 oxen cost $1445. 27. If 224 acres cost $5600, what did 1 acre cost? 28. If 864 sheep cost $3456, what was the cost of 1 sheep? GREATEST COMMON DIVISOR. 86. A Divisor of a number is a number that will exactly divide it. A Factor of a number is a divisor of it. A prime factor cannot be resolved into factors other than itself and unity. 87. A Common Divisor of two or more numbers is an exact divisor of each of them. 88. The Greatest Common Divisor of two or more numbers is the greatest number that will exactly divide each of them. FUNDAMENTAL PRINCIPLE. The Greatest Common Divisor of several numbers is the product of all the factors common to the several numbers. First. If this product contains any factor not found in either of the numbers, it will not exactly divide that number. Second. If this product does not contain all the factors common to the several numbers, it will not be the greatest common divisor of them. FIRST METHOD. 89. To find the Greatest Common Divisor of several numbers by factoring. Example 1. - Find the Greatest Common Divisor of 12, 32, 44, and 56. Solution. -The prime factors of 12 are 2, 2, and 3. The prime factors of 32 are 2, 2, 2, 2, and 2. The prime factors of 56 are 2, 2, 2, and 7. Since 3 is not a factor of 32, 44, and 56, it can form no part of the Greatest Common Divisor. For a similar reason 7 and 11 can form no part of the Greatest Common Divisor. Since 2, taken twice, is the only factor common to all the numbers, their product, 4, is the greatest Common Divisor sought. Example 2.- Find the Greatest Common Divisor of 36, 48, 84, and 96. Solution. - The prime factors of 36 are 2, 2, 3, and 3. The prime factors of 96 are 2, 2, 2, 2, 2, and 3. The only factors common to these several numbers are 2, 2, and 3. Hence, 12, their product, is the Greatest Common Divisor. 3. Find the Greatest Common Divisor of 24, 38, 56, and 74. 4. Find the Greatest Common Divisor of 23, 46, 69, and 115. 5. Find the Greatest Common Divisor of 21, 63, 84, and 105. 6. Find the Greatest Common Divisor of 51, 85, 119, and 153. 7. Find the Greatest Common Divisor of 39, 91, 169, and 195. 8. Find the Greatest Common Divisor of 45, 81, 108, and 189. 9. Find the Greatest Common Divisor of 144, 360, 432, and 648. 10. Find the Greatest Common Divisor of 431, 341, and 143. SECOND METHOD. 90. This Method of finding the Greatest Common Divisor of two or more numbers depends on the following principles : First Principle. A Divisor of a number is a divisor of any multiple of that number. Demonstration. - Let a be any number, and b a divisor of it. Then 2a, 3a, 4a, na, represent multiples of a. It is evident that if b is a divisor of a, that is, if b is contained in a a certain number of times, it must be contained in 2a, 3a, 4a; na, twice, three times, four times, n times as many times. Second Principle. A Divisor of two or more numbers is also a divisor of their sum. Demonstration. - Let a be any number, and b a divisor of it. Then by FIRST PRINCIPLE b is a divisor of 2a, 3a, 4a, na. But b, being a divisor of a, and also of 2a, is a divisor of 3a, their sum. It is also a divisor of a plus 3a, or 4a, of a plus 4a, or 5a. Generally, of na plus a, or, a (n+1) Third Principle. A Divisor of two numbers is also a divisor of their difference. Demonstration. - Let a be any number, and b a divisor of it. Then, by FIRST PRINCIPLE, a is a divisor of 4a and 7a. But a is also a divisor of 3a, their difference. Example 1. - Find the Greatest Common Divisor of 2043 and 5221. Explanation. be a divisor of 4086. 4th Divisor 227)908 (4 908 A divisor of 2043 must, by Principle First, The divisor sought, then, is a divisor of 4086 and 5221, and must, by Principle Third, be a divisor of the difference, 1135. Since the divisor sought is a divisor of 1135 and 2043, it must, by the Third Principle, be a divisor of 908, their difference. Since the divisor sought is a divisor of 908 and also a divisor of 1135, it must, by Principle Third, be a divisor of their difference, which is 227. Proof. Since the divisor sought is a divisor of 227, it must, by the First Principle, be a divisor of any multiple of 227, that is, of 908. |