3. What are the prime factors of 50? 56? 58? 60? 64? 66? 68? 70 ? *72? 4. What are the prime factors of 76? 78? 80? 82? 84? 86? 88? 90? 5. What are the prime factors of 100? 102? 104? 275? 960? 472? 160? 836? 6. What are the prime factors of 105? 106? 108? 110? 115? 116? 120? 125? 1125? 360 ? NOTE.-1 he prime factors, when the numbers are small, may generally be seen by inspection. The teacher can easily increase the number of examples. 111. To find the prime factors common to two or more composite numbers. 1. What are the common prime factors of 90, 120, and 150? 60.. 75 2) 90.. 120. . 150 ANALYSIS.-It is plain that 2 is an exact divisor of all the numbers: hence, it is a prime factor of them. Since 3 is an exact divisor of the quotients, it is a prime factor of them; and since 5 will divide the second set of quotients, it is a prime factor. The quotients 3, 4, and 5 have no exact divisor; therefore, 2, 3, and 5 are all the common prime factors: hence, 3.. 4 25 5 . The common prime factors of two or more numbers, are the exact divisors common to them all. Examples. 1. What are the prime factors common to 150, 210, and 270? 2. What are the prime factors common to 42, 126, and 168? 3. What are the prime factors common to 105, 315, and 525? 4. What are the prime factors common to 84, 126, and 210? 5. What are the prime factors common to 168, 256, 410, and 820? 111. What is the rule for finding the prime factors common to two or more composite numbers? CANCELLATION. 112. CANCELLATION is a method of shortening Arithmetical operations by omitting or cancelling common factors. 1. Divide 36 by 18. First, 36 = 9×4; and 18 = 9 × 2. ANALYSIS.-Thirty-six divided by 18 is equal to 9 x 4 divided by 9 x 2: by cancelling, or striking out the 9's, we have 4 divided by 2, which is equal to 2. NOTE. The figures cancelled are slightly crossed. 113. Principles-Operations-and Rule. The operations, in cancellation, depend on two principles: 1. The cancelling of a factor, in any number, is equivalent to dividing the number by that factor. 2. If the dividend and divisor be both divided by the same number, 'the quotient will not be changed. 2. In 72 times 25, how many times 45? ANALYSIS.-We see that 9 is a factor of 72 and 45. Divide by 9, and write the quotient 8 over 72, and the quotient 5 below 45. Again, 5 is a factor of 25 and 5. Divide 25 by 5, and write the quo tient 5 over 25. Dividing 5 by 5, reduces the divisor to 1: hence, the true quotient is 40. Rule. I. Resolve the dividend and divisor into their prime factors or conceive them to be so resolved: II. Cancel the common factors, and then divide the product 112. What is Cancellation? 113. On what principles does the operation of cancellation depend? What is the rule for the operation? of the remaining factors of the dividend by the product of the remaining factors of the divisor. NOTES.-1. Since every factor is cancelled by division, the quotient 1 always takes the place of the cancelled factor. 2. If one of the numbers contains a factor equal to the product of two or more factors of the other, all such factors may be cancelled. 3. If the product of two or more factors of the dividend is equal to the product of two or more factors of the divisor, such factors may be cancelled. Examples. 1. What is the quotient of 2x4x8x13x7x16, divided by 26×14×8? 2. What is the quotient of 42 × 3 × 25 x 12, divided by 28 × 4 × 15×6? 3. What is the quotient of 125 × 60 × 24 × 42, divided by 25 x 120 x 36x5? 4. How many times is 11 x 39 × 7 × 2 contained in 44 × 18 x 26 x 14? 5. What is the quotient of 8 times 240 multiplied by 5 times 114, divided by 24 times 57 multiplied by 6 times 15? 6. What is the value of (22+8+16) × (18+10+21) divided by (9+5+7) × (15+8) ? 7. Divide (140 + 86 −34) × (107—19) by (237 — 141) × (17+20-15)? 8. Divide [12×5−2×9] × (42+30) by (5×8) × (2 × 9) × (10+17)? 9. What is the quotient of 240 × 441 × 16 divided by 175 × 56 × 27? 10. What is the quotient of 64 times 840 multiplied by 9 times 124, divided by 32 times 560 multiplied by 4 times 31? 11. How many dozen of eggs, worth 14 cents a dozen, must be given for 18 pounds of sugar, worth 7 cents a pound? 12. A dairyman sold 5 cheeses, each weighing 40 pounds, at 9 cents a pound: how many pounds of tea, worth 50 cents a pound, must he receive for the cheeses? 13. Bought 12 yards of cloth, at $1.84 a yard, and paid for it in potatoes at 48 cents a bushel: how many bushels of potatoes will pay for the cloth? 14. How many firkins of butter, each containing 56 pounds, at 25 cents a pound, will pay for 4 barrels of sugar, each weighing 175 pounds, at 8 cents a pound? 15. A man bought 10 cords of wood, at 20 shillings a cord, and paid in labor at 12 shillings a day: how many days did he labor? 16. How many pieces of cloth, each containing 36 yards, at $3.50 a yard, must be given for 96 barrels of flour, at $10.50 a barrel ? 17. A farmer exchanged 492 bushels of wheat, worth $1.84 a bushel, for an equal number of bushels of barley, at 87 cents a bushel; of corn, at 60 cents a bushel; and of oats, at 45 cents. a bushel how many bushels of each did he receive? 18. How many barrels of flour, worth $7 a barrel, must be given for 250 bushels of oats, at 42 cents a bushel? 19. If 48 acres of land produce 2484 bushels of corn, how many bushels will 120 acres produce? 20. A man worked 12 days, at 9 shillings a day, and received in payment wheat at 16 shillings a bushel: how many bushels did he receive? 21. A grocer sold 6 hams, each weighing 14 pounds, at 10 cents a pound, and received in payment apples, at 48 cents a bushel how many bushels of apples did he receive? 22. How long will it take a man, travelling 36 miles a day, to go the same distance that another man travelled in 15 days, at the rate of 27 miles a day? 23. A man took four loads of apples to market, each load containing 12 barrels, and each barrel 3 bushels. He sold them at 45 cents a bushel, and received in payment a number of boxes of tea, each box containing 20 pounds, worth 72 cents a pound: how many boxes of tea did he receive? LEAST COMMON MULTIPLE. 114. A MULTIPLE of a number is any product of which the number is a factor; hence, any multiple of a number is exactly divisible by the number itself. 115. A COMMON MULTIPLE of two or more numbers is any number which each will divide without a remainder. 116. THE LEAST COMMON MULTIPLE of two or more numbers is the least number which they will separately divide without a remainder. 117. Principles-Operations-and Rule. 1. Any divisible number, is divisible by any prime factor of the exact divisor. 2. If a number has several exact divisors, it will be divisible by all their prime factors. 3. Hence, the question of finding the least common multiple of sev eral numbers is reduced to finding a number which shall contain all their prime factors, and none others. 1. What is the least common multiple of 6, 12, and 18? OPERATION. 12 18 2) 6 6 9 2 3 1 ANALYSIS. Having placed the given numbers in a line, if we divide by 2, we find the quotients 3, 6 and 9; hence, 2 is a prime factor of all the numbers. Dividing by 3, we find that 3 is a prime factor of the quotients 3, 6, and 9; and hence, the quotients 2 and 3 are prime factors of 12 and 18; therefore, the prime factors of all the numbers are 2, 3, 2 and 3; and their product, 36, is the least common multiple. 2 × 3 × 2 × 3 = 36 114. What is a multiple of a number?-115. What is a common multiple of two or more numbers?-116. What is the least common multiple of two or more numbers? 117. What is the first principle on which the operation for finding the least common multiple depends? What is the second? What is the third? Give the rule for finding the least common multiple. |