Examples. 1. What number is equal to the product of 36 and 13, divided by the product of 4 and 9? ANALYSIS.-We see that 4 times 9 are equal to 36; therefore, we cancel the 36, and the 4 and 9. 2. Divide 960 by 480. OPERATION. 36 × 13 == = 13. 1 OPERATION. 960 96 8 480 = = 2. 48 4 ANALYSIS.-We see that 10 is a common factor; then 12, then 4. We may divide mentally, by the common factors, and place the results at the right. 3. Divide the product of 6×7×9×11, by 2×3×7×3 × 21. 4. Divide the product of 4×14×16×24, by 7×8×32 X 12. 5. Divide the product of 5 × 11 × 9 × 7 × 15×6, by 30 3×21×3×5. 6. Divide 285120 by 5184. 7. Divide 5080320 by 635040. 8. How much calico, at 25 cents a yard, must be given for 8700 cents? 9. How many yards of cloth, at 46 cents a yard, can be bought for 2116 cents? 10. How much molasses, at 42 cents a gallon, can be bought for 1512 cents? 11. In a certain operation, the numbers 24, 28, 32, 49, 81, are to be multiplied together, and the product divided by 8×4×79×6: what is the result? 12. How many pounds of butter, worth 15 cents a pound, may be bought for 25 pounds of tea, at 48 cents a pound? 13. How many bushels of oats, at 42 cents a bushel, must be given for 3 boxes of raisins, each containing 26 pounds, at 14 cents a pound? 14. A man buys 2 pieces of cotton cloth, each containing 33 yards, at 11 cents a yard, and pays for it in butter at 18 cents a pound: how many pounds of butter did he give? 15. If sugar can be bought for 7 cents a pound, how many bushels of oats, at 42 cents a bushel, must I give for 56 pounds? 16. Bought 48 yards of cloth, at 125 cents a yard: how many bushels of potatoes are required to pay for it, at 150 cents a bushel ? 17. Mr. Butcher sold 342 pounds of beef, at 6 cents a pound, and received his pay in molasses at 36 cents a gallon: how many gallons did he receive? 18. Mr. Farmer sold 1263 pounds of wool, at 5 cents a pound, and took his pay in cloth at 441 cents a yard: how many yards did he take? 19. How many firkins of butter, each containing 56 pounds, at 18 cents a pound, must be given for 3 barrels of sugar, each containing 200 pounds, at 9 cents a pound? 20. How many boxes of tea, each containing 24 pounds, worth 5 shillings a pound, must be given for 4 bins of wheat, each containing 145 bushels, at 12 shillings a bushel? 21. A. worked for B. 8 days, at 6 shillings a day, for which he received 12 bushels of corn: how much was the corn worth a bushel ? 22. Bought 15 barrels of apples, each containing 2 bushels, at the rate of 3. shillings a bushel: how many cheeses, each weighing 30 pounds, at 1 shilling a pound, will pay for the apples? Least Common Multiple. 91. A MULTIPLE of a number, is the product of that number by some other number. Thus, the dividend is a multiple of the divisor or quotient. 92. A COMMON MULTIPLE of two or more numbers, is a number exactly divisible by each of them. 93. The LEAST COMMON MULTIPLE of two or more numbers, is the least number which is divisible by each 91. What is a multiple of a number? 92. What is a common multiple of two or more numbers? 93. What is the least common multiple of two or more numbers? of them. Thus, 18 is the least common multiple of 2, 6, and 9. NOTES.-1. If a division is exact, the dividend may be resolved into two factors, one of which will be the divisor, and the other the quotient. 2. If the divisor be resolved into its prime factors, the corresponding factor of the dividend may be resolved into the same factors: hence, the dividend will contain every factor of an exact divisor. 3. The question of finding the least common multiple of several numbers, is, therefore, reduced to finding a number which shall contain all the prime factors of the given numbers, and none others. 1. Find the prime factors and least common multiple of 6, 12, and 18. OPERATION. 2) 6. 12. 18 3) 3 6 9 1 2 3 ANALYSIS.-Write the numbers in a line, and then divide them and the quotients which follow, by any prime number that will exactly divide two or more of them, until quotients are found which are prime with each other. It is plain, that the divisors, 2 and 3, are prime factors of 6; 2, 3, and the quotient 2, of 12; and 2, 3, and the quotient 3, of 18: hence, the prime factors are 2, 3, 2, and 3, and their product, 2 × 3 × 2 × 3 = the least common multiple. 94. Hence, to find the least common multiple, Rule. = 36, I. Write the numbers in a line, and divide by any prime number that will exactly divide any two of them, and write down the quotients, and the undivided numbers. II. Divide as before, until there is no exact divisor of any two of the quotients: the product of the divisors and the final quotients, will be the least common multiple. Examples. 1. Find the least common multiple of 3, 4, and 8. 2. Find the least common multiple of 3, 8, and 9. 3. Find the least common multiple of 6, 7, 8, and 10. 4. Find the least common multiple of 21 and 49. 5. Find the least common multiple of 2, 6. Find the least common multiple of 4, 7, 5, 6, and 8. 14, 28, and 98. 7. Find the least common multiple of 13 and 6. 8. Find the least common multiple of 12, 4, and 7. 9. Find the least common multiple of 6, 9, 4, 14, and 16 10. Find the least common multiple of 13, 12, and 4. Greatest Common Divisor. 95. A COMMON DIVISOR of two or more numbers, is any number that will divide each of them without a remainder. 96. The GREATEST COMMON DIVISOR of two or more numbers, is the greatest number that will divide each of them without a remainder. 97. Two numbers are prime with each other, when they have no common divisor. CASE I. 98. To find the greatest common divisor of two or more numbers, when the numbers are small. Since an exact divisor is a factor, the greatest common divisor of the given numbers, will be their greatest common factor: Hence, Rule. Find the prime factors common to all the numbers (Art. 87), and their product will be the greatest common divisor. 94. What is the rule for finding the least common multiple? 95. What is a common divisor of two or more numbers? 96. What is the greatest common divisor of two or more numbers? 97. When are two numbers prime with each other? 98. How do you find the greatest common divisor, when the numbers are small? Examples. 1. What is the greatest common divisor of 24 and 30? 2. What is the greatest common divisor of 9 and 18? 3. What is the greatest common divisor of 6, 12, and 30? 4. What is the greatest common divisor of 15, 25, and 30? 5. What is the greatest common divisor of 12, 18, and 72? 6. What is the greatest common divisor of 25, 35, and 70? 7. What is the greatest common divisor of 28, 42, and 70? 8. What is the greatest common divisor of 84, 126, and 210? CASE II. 99. To find the greatest common divisor, when the numbers are large. The operation of finding the common divisor, depends on the following principles: ILLUSTRATION. 24168 1. Any number which will exactly divide the difference of two numbers, and one of them, will exactly divide the other; else, we should have a whole number equal to a fraction, which is impossible. 2. Any number that will exactly divide another, will divide. any multiple of that other; because, the first dividend is a factor of the multiple, and any number which will divide a factor, will divide the multiple. OPERATION. 50 1. What is the greatest common divisor of 25 and 70? ANALYSIS.-Divide the greater number, 70, by the less, 25; we find a quotient 2, and a remainder 20. Then divide the divisor 25 by the remainder 20; the quotient is 1, and the remainder 5. Then divide the divisor 20 by the remainder 5; the quotient is 4, and the division exact. Now, the remainder 5, exactly divides itself and 20; hence, by the first prin 25) 70 (2 20) 25 (1 20 5) 20 (4 20 ciple, it will exactly divide 25. Since 5 divides 25, it will, by the second principle, divide 50, a multiple of 25; but since it |