94. Whenever the dividend and divisor are each composite numbers, the factors common to both may first be rejected without altering the final result. 2. What is the quotient of 24 48? OPERATION. 21 × 56 4 × 6 × 7 X $ 7 × 48 (87, Prin. III.) times 56 divided by 7 times 7 × 6 × 8 eration to be performed by writing the numbers which constitute the dividend above a line, and those which constitute the divisor below it. Instead of multiplying 24 by 56, in the dividend, we resolve 24 into the factors 4 and 6, and 56 into the factors 7 and 8; and 48 in the divisor into the factors 6 and 8. We next cancel the factors 6, 7, and 8, which are common to the dividend and divisor, and we have left the factor 4 in the dividend, which is the quotient. NOTE. When all the factors or numbers in the dividend are canceled, 1 should be retained. 95. If any two numbers, one in the dividend and one in the divisor, contain a common factor, we may reject that factor. 3. In 54 times 77, how many times 63 ? ANALYSIS. In this example we see that 9 will divide 54 and 63; so we reject 9 as a factor of 54, and retain the factor 6, and also as a factor of 63, and retain the factor 7. the divisor, and 77 in both numbers by 7, 1 will be retained in the divisor, and 11 in the dividend. Finally, the product of 6 X 1166, the quotient. Again, 7 will divide 7 in the dividend. Dividing 4. Divide 25 X 16 X 12 by 10 X 4 X 6 X 7. and we have remaining the factor 7 in the divisor, and the factors 5 and 4 in the dividend. Completing the work, we have 20=2§, Ans. From the preceding examples and illustrations we derive the following RULE. I. Write the numbers composing the dividend above a horizontal line, and the numbers composing the divisor below it. II. Cancel all the factors common to both dividend and divisor. III. Divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor, and the result will be the quotient. NOTES. 1. Rejecting a factor from any number is dividing the number by that factor. 2. When a factor is canceled, the unit, 1, is supposed to take its place. 3. One factor in the dividend will cancel only one equal factor in the divisor. 4. If all the factors or numbers of the divisor are canceled, the product of the remaining factors of the dividend will be the quotient. 5. By many it is thought more convenient to write the factors of the dividend on the right of a vertical line, and the factors of the divisor on the left. EXAMPLES FOR PRACTICE. 1. What is the quotient of 16 × 5 × 4 divided by 20 × 8? 2. Divide the product of 120 X 44 X 6 X 7 by 72 X 33 X 14. Rule, first step? Second? Third? What is the effect of rejecting a factor? What is the quotient when all the factors in the divisor are canceled? 3. Divide the product of 33 × 35 × 28 by 11 × 15 X 14. Ans. 14. 4. What is the quotient of 21 × 11 × 26 divided by 14 X 13? Ans. 33. 28, and 5, by and give the Ans. 94. 5. Divide the product of the numbers 48, 72, the product of the numbers 84, 15, 7, and 6, result. 6. Divide 140 × 39 × 13 × 7 by 30 × 7 × 26 X 21. Ans. 4. Ans. 10. 7. What is the quotient of 66 × 9 × 18 × 5 divided by 22 X 6 X 40? 8. Divide the product of 200 × 36 × 30 × 21 by 270 X 40 X 15 X 14. 9. Multiply 240 by 56, and divide the product by 60 multiplied by 28. 10. The product of the numbers 18, 6, 4, and divided by the product of the numbers 4, 9, 3, 7, is the result? Ans. 2. Ans. 8. 42 is to be what 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? Ans. 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. NOTE. 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 ? Ans. 2. 6..10 OPERATION. 2 3 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. 5 2. What is the greatest common divisor of 42, 63, and 105? What is a common divisor? The greatest common divisor? A common measure? The greatest common measure? What is Case I? Give analysis. 3 OPERATION. 7 14.. 21 .. 35 2.. 3 5 .. 3 X 721, 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, and 280? 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. EXAMPLES FOR PRACTICE. 1. What is the greatest common divisor of 12, 36, 60, 72? Ans. 12. 2. What is the greatest common divisor of 18, 24, 30, 36, 42? Rule, first step? Second? Third? Ans. 6. |