sum as there are nines in the divisor; the figures on the left hand constitute the quotient, and those cut off the remainder. The figures extended without the line show the reason of carrying the unit when necessary. Q. When the divisor is above 12, how do you divide? Q. How do you divide, when the divisor is a composite number? Q. How do you prove division? CASE 5. § 17. To resolve any composite number into its prime factors. RULE. Divide the number by any prime number which will divide it without a remainder; then divide the quotient in the same way, and so continue until a quotient is obtained which is a prime number. Then the successive divisors together with the last quotient will form the prime factors required. The reason of this rule may be found in the fact, that all numbers which are not prime, are composed of prime factors. For, all numbers which are not prime are composite, and may be therefore separated into two or more factors. If these factors are not prime, they can again be separated into other factors, and thus the decomposition can be continued until all the factors are prime. E 2. Resolve 450 into its prime factors. 3. Resolve 786 into its prime factors. 4. Resolve 897 into its prime factors. 5. Resolve 1998 into its prime factors. 6. Resolve 2799 into its prime factors. 7. Resolve 3976 into its prime factors. 8. Resolve 14950 into its prime factors. 9. Resolve 32368 into its prime factors. 10. Resolve 786542 into its prime factors. 11. Resolve 98765432 into its prime factors. Q. What are prime factors? Q. How do you resolve a number into prime factors? MISCELLANEOUS EXERCISES. 1. Divide 29629608 by 2. 2. Divide 4938268 by 4. 3. Divide 269867904 by 12. 4. Divide 503700480 by 15. 5. Divide 844439040 by 18. 6. Divide 4552933760 by 32. 7. Divide 7125880320 by 37. 8. Divide 103703040 by 28. 9. Divide 4320960 by 35. 10. Divide 99550080 by 84. 11. How many times is 72 contained in 7167605760 ? 16. Divide 2922839479575 by 23675. 18 Divide 168434689825269 by 1364321 19. Divide 596566908151488 by 4832192. 20. Divide 7927200051319269 by 64210321. 21. Multiply 13717421 by 72, and divide their product by 8. 22. Multiply 13717421 by 1827, and divide their product by 203. 23. An army of 19000 men having plundered a city of 266000 dollars. How much must each man have? 24. There was a certain number of men concerned in the payment of 1272 dollars, and each one paid three dollars. What was the number of men? 25. If 2233 barrels of flour were divided among eleven persons, how much would each one receive? 26. The guardians of the poor have 2214 cords of wood to divide among 738 persons. How many cords will each one receive? 27. France has 32000000 inhabitants. The number of square miles in her territory is 154000. How many persons has she to each mile? 28. Eight children agreed to divide equally the estate left by their father. The sum divided was 67456 dollars. What was the share of each? 29. In planting a large orchard a farmer made ninety rows. How many trees should he put in each row to make up 4320, the whole number of trees planted? 30. Sixty-six German emigrants bought 42240 acres of land in the state of Ohio, which they resolve to divide equally. How many acres will each receive? 31. In starting upon a journey of 2376 miles, a traveller resolves to ride 44 miles a day. How many days, at that rate, will it take him to finish his journey? 32. From New York to Greece is 4800 miles. How many miles must a ship sail in a day to make the passage in 24 days? 33. In the year 1755, a vessel sailed from France to Canada, 2800 miles, in four weeks. How many miles did she sail in a week? 34. How many days would it require to sail from Scotland to Labrador, a distance of 2100 miles, at the rate of 150 miles each day? 35. From Norway to Greenland is 1800 miles. From Chili to New Zealand is 5800 miles. If a ship start from Norway and another from Chili at the same time, one for Greenland, the other for New Zealand, and each sail at the rate of 100 miles a day, how much longer will one vessel be in reaching New Zealand than the other in making Greenland? 36. In the state of Virginia there are 1239797 inhabitants, and 64000 square miles. How many persons are there to each square mile? 37. Ninety-nine labourers pick in one year 12474 pounds of cotton. How many pounds is that for each? 38. In a regiment consisting of 1000 men, there were 50 officers, how many men to an officer? 39. The value of the machinery manufactured annually in the United States, is about 10985845 dollars. The number of men employed is 13001. What amount of machinery is produced by each man? 40. The number of smelting houses for the manufacture of lead is 120. The number of pounds of lead produced is about 31239480. How many pounds would that give for each smelting house? 41. The number of men employed in the manufacture of hardware and cutlery is about 5500. The value of the manufacture may be estimated at 6451500 dollars. How much in that case does each man produce? 42. The number of wholesale commercial and commission houses in the city and county of Philadelphia is 149; that of the stores for retailing dry goods, groceries, &c. is about 2086. How many of the latter are there for each of the former? 43. The number of forges, furnaces, &c. for the manufacture of wrought and cast iron in the United States, is about 1600. The capital invested in the whole of them is nearly 20433600 dollars. What would be the capital of each were it equally divided? 44. The total number of houses annually erected in the United States, is showed by the census of 1840, to be 54113. The erection of these costs about 41911609 dollars. How much is that for each? 45. By the same census, the number of powder mills in the United States is 137. The number of pounds of powder manufactured annually, is 8977336. What quantity would that give for each, allowing every mill to produce an equal number of pounds? COMPOUND ADDITION. § 18. Compound Addition is the addition of several numbers of different denominations. RULE. 1. Place all the given numbers of the same denomination directly under each other. 2. Add up the figures in the lowest denomination, and divide the sum by as many of that denomination as will make one of the next greater. 3. Place the remainder under its column, and carry the quotient to the next; which add up in the same manner as before. 4. Proceed thus through all the denominations to the highest, whose sum, together with the several remainders, will be the answer required. The method of proof is the same as in addition of integers. Q. What is compound addition? Q. How are the numbers placed to be added? Q. Where must you begin to add? Q. If the amount of the column added is more than will make one of the next higher denomination, what do you do? Q. If the amount of the column is less than one of the next higher denomination, how do you proceed? A. Set down the whole amount of the column. FEDERAL MONEY. § 19. Federal money is the currency of the United States. Q. What are the denominations of federal money? A. Eagles, dollars, dimes, cents, and mills. Repeat the table. 1 eagle, E. The coins of the United States are of three kinds; gold, silver, and copper. |
