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 × 12, divided by 28 x 4 x15x6?

3. What is the quotient of 125 × 60 × 24 × 42, divided by 25 × 120 × 36 × 5?

4. How many times is 11 x 39 x 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 (5x8) × (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 oxes of tea, each box containing 20 pounds, worth 72 cents 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.

2) 6
3)3.

1

12

6

2

18

9

3

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.

Rule.

I. Place the numbers on the same line, and divide by any PRIME number that will exactly divide two or more of them, and set down, in a line below, the quotients and the undivided numbers:

II. Then divide as before, until there is no prime number greater than 1 that will exactly divide any two of them:

III. Then multiply together the divisors and the numbers of the lower line, and their product will be the least common multiple.

NOTE. If the numbers have no common prime factor, their product will be their least common multiple.

Examples.

1. What is the least common multiple of 4, 9, 10, 15, 18, 20, 21?

2. What is the least common multiple of 8, 9, 10, 12, 25, 32, 75, 80?

3. What is the least common multiple of 1, 2, 3, 4, 5, 6, 7, 9?

4. What is the least common multiple of 9, 16, 42, 63, 21, 14, 72?

5. What is the least common multiple of 7, 15, 21, 28, 35, 100, 125 ?

6. What is the least common multiple of 15, 16, 18, 20, 24, 25, 27, 30?

7. What is the least common multiple of 9, 18, 27, 36, 45, 54? 8. What is the least common multiple of 4, 10, 14, 15, 21? 9. What is the least common multiple of 7, 14, 16, 21, 24? 10. What is the least common multiple of 49, 14, 84, 168, 98?

11. A can dig 9 rods of ditch in a day; B, 12 rods in a day; and C, 16 rods in a day: what is the smallest number of rods that would afford exact days of labor to each, working alone? In what time would each do the whole work?

12. A blacksmith employed 4 classes of workmen, at $15, $16, $21, and $24 per month, for each man respectively, paying to each class the same amount of wages. Required the least amount that will pay either class for 1 month; also, the number of men in each class?

13. A farmer has a number of bags containing 2 bushels each; of barrels, containing 3 bushels each; of boxes, containing 7 bushels each; and of hogsheads, containing 15 bushels each: what is the smallest quantity of wheat that would fill each an exact number of times, and how many times would that quantity fill each?

14. Four persons start from the same point to travel round a circuit of 300 miles in circumference. A goes 15 miles a day; B, 20 miles; C, 25 miles; and D, 30 miles a day. How many days must they travel before they will all come together again at the same point, and how many times will each have gone round? NOTE.-First find the number of days that it will take each to travel round the circuit.

GREATEST COMMON DIVISOR.

118. A COMMON DIVISOR of two or more numbers, is any number that will divide each of them without a remainder; hence, it is always a common factor of the numbers.

119. THE GREATEST COMMON DIVISOR of two or more numbers, is the greatest number that will divide each of them without a remainder; hence, it is their greatest common factor.

120. Two numbers are said to be prime to each other, when they have no common divisor.

NOTE. Since 1 will divide every number, it is not reckoned among the common divisors.

118. What is a common divisor?-119. What is the greatest common divisor?

120. When are two numbers said to be prime to each other?