3. Divide the product of 33 × 35 × 28 by 11 × 15 × 14. Ans. 14. 4. What is the quotient of 21 × 11 × 26 divided by 14 × 13? Ans. 33. 5. Divide the product of the numbers 48, 72, 28, and 5, by the product of the numbers 84, 15, 7, and the result. 6, and give Ans. 94. Ans. 41. Ans. 101. 8. Divide the product of 200 x 36 x 30 x 21 by 270 x 40 x 15 x 14. Ans. 2. 7. What is the quotient of 66 × 9 × 18 × 5 divided by 22 × 6 × 40. 9. Multiply 240 by 56, and divide the product by 60 mui tiplied by 28. Ans. 8. 10. The product of the numbers 18, 6, 4, and 42 is to be divided by the product of the numbers 4, 9, 3, what is the result? 7 and 6; Ans. 4. 11. How many tons of hay, at 12 dollars a ton, must be given for 30 cords of wood, at 4 dollars a cord? A. 10 tons. 12. How many firkins of butter, each containing 56 pounds, at 13 cents a pound, must be given for 4 barrels of sugar, each containing 182 pounds, at 6 cents a pound? Ans. 6 firkins. 13. A tailor bought 5 pieces of cloth, each piece containing 24 yards, at 3 dollars a yard. How many suits of clothes, at 18 dollars a suit, must be made from the cloth to pay for it? Ans. 20 suits. 14. How many days' work, at 75 cents a day, will pay for 115 bushels of corn, at 50 cents a bushel? Ans. 763 days. GREATEST COMMON DIVISOR. 96. A Common Divisor of two or more numbers is a number that will exactly divide each of them. 97. The Greatest Common Divisor of two or more numbers is the greatest number that will exactly divide each of them. Numbers prime to each other are such as have no common divisor. A common divisor is sometimes called a Common Measure; and the greatest common divisor, the Greatest Common Measure. CASE I. 98. When the numbers are readily factored. 1. What is the greatest common divisor of 6 and 10? OPERATION. 26.. 10 3.. 5 Ans. 2. ANALYSIS. We readily find by inspection that 2 will divide both the given numbers; hence 2 is a common divisor; and since the quotients 3 and 5 have no common factor, but are prime to each other, the common divisor, 2, must be the greatest common divisor. 2. What is the greatest common divisor of 42, 63, and 105? What is a common divisor? The greatest common divisor? A cominon measure? The greatest common measure? What is Case I.? Give analysis. 3 OPERATION. 42.. 63.. 105 14 21 35 2.. 3.. 5 3x7=21, Ans. ANALYSIS. We observe that 3 will exactly divide each of the given numbers, and that 7 will exactly divide each of the resulting quotients. Hence, each of the given numbers can be exactly divided by 3 times 7; and these numbers must be component factors of the greatest common divisor. Now, if there were any other component factor of the greatest common divisor, the quotients, 2, 3, 5, would be exactly divisible by it. But these quotients are prime to each other. Hence 3 and 7 are all the component factors of the greatest common divisor sought. 3. What is the greatest common divisor of 28, 140, 280? and ANALYSIS. We first divide by 4; then the quotients by 7. The resulting quotients, 1, 5, and 10, are prime to each other. Hence 4 and 7 are all the component factors of the greatest common divisor. From these examples and analyses we derive the following RULE. I. Write the numbers in a line, with a vertical line at the left, and divide by any factor common to all the numbers. II. Divide the quotients in like manner, and continue the division till a set of quotients is obtained that have no common factor. III. Multiply all the divisors together, and the product will be the greatest common divisor sought. 1. What is the greatest common divisor of 12, 36, 60, 727 Ans. 12. 2. What is the greatest common divisor of 18, 24, 30, 36, 42? Rule, first step? Second? Third? Ans. 6. 3. What is the greatest common divisor of 72, 120, 240, 384 ? Ans. 24. 4. What is the greatest common divisor of 26, 126, 72, 216? Ans. 18. 5. What is the greatest common divisor of 42 and 112? Ans. 14. 6. What is the greatest common divisor of 32, 80, and 256? Ans. 16. 7. What is the greatest common divisor of 210, 280, 350, 630, and 840 ? Ans. 70. 8. What is the greatest common divisor of 300, 525, 225, and 375? Ans. 75. 9. What is the greatest common divisor of 252, 630, 1134, and 1386? Ans. 126. 10. What is the greatest common divisor of 96 and 544? Ans. 32. 11. What is the greatest common divisor of 468 and 1184? Ans. 4. 12. What is the greatest common divisor of 200, 625, 150? CASE II. and Ans. 25. 99. When the numbers can not be readily factored. As the analysis of the method under this case depends upon three properties of numbers which have not been introduced, we present them in this place. I. An exact divisor divides any number of times its divi dend. II. A common divisor of two numbers is an exact diviso of their sum. III. A common divisor of two numbers is an exact divisor of their difference. What is Case II.? What is the first principle upon which it is founded? Second? Third? 1. What is the greatest common divisor of 84 and 203? OPERATION. 203 81 2 168 70 2 35 14 2 28 14 2 , Ans. ANALYSIS. Draw two vertical lines, and place the larger number on the right, and the smaller number on the left, one line lower down. Then divide 203, the larger number, by 84, the smaller, and write 2, the quotient, between the verticals, the product, 168, opposite, under the greater number, and the remainder, 35, below. Next divide 84 by this remainder, writing the quotient, 2, between the verticals, the product, 70, on the left, and the new remainder, 14, below the 70. Again divide the last divisor, 35, by 14, and obtain 2 for a quotient, 28 for a product, and 7 for a remainder, all of which we write in the same order as in the former steps. Finally, divide the last divisor, 14, by the last remainder, 7, and we have no remainder. 7, the last divisor, is the greatest common divisor of the given numbers. 0 In order to show that the last divisor in such a process is the greatest common divisor, we will first trace the work in the reverse order, as indicated by the arrow line below. 7 divides the 14, as proved by the last division; it will also divide two times 14, or 28, (I.) Now, as 7 divides both itself and 28, it will divide 35, their sum, (II.) It will also divide 2 times 35, or 70, (I;) and since it is a common divisor of 70 and 14, it must divide their sum, 84, which is one of the given numbers, (II.) It will also divide 2 times 84, or 168, (I;) and since it is a common divisor of 168 and 35, it must divide their sum, 203, the larger number, (II.) Hence 7 is a common divisor of the given numbers. Again, tracing the work in the direct order, as indicated below, we Give analysis. |