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16. I have three rooms, the first 15 feet wide, the second 24 feet wide, and the third 36 feet wide: what is the width of the widest carpet that will fit either room ?

17. Mr. Emmons wishes to divide his three farms of 324 acres, 486 acres, and 4,293 acres respectively, into tracts of equal size : what is the greatest number of acres each tract may contain ?

18. Mr. Jones shipped by rail in equal car-loads four lots of wood, the first containing 84, the second 96, the third 108, and the fourth 120 cords : how many cords in each car-load, the loads being the largest possible ?

19. Illustrate by an original problem each method of finding the greatest common divisor.

Multiples.

ORAL EXERCISES.

ART. 100.-1. Of what numbers is 8 a multiple? 12 ? 15 ? 20 ? 24 ? 25 ? 32 ? 36 ? 48 ?

2. Name three multiples of 4; of 5; of 8; of 9; of 12; of 14 ; of 11; of 13; of 15.

3. Name a common multiple of 3 and 4; of 4 and 5; of 6 and 7; of 5 and 8.

4. Name a common multiple of 2, 3 and 4; of 3, 5 and 6; of 5, 6 and 8; of 2, 5 and 10.

5. Name the least common multiple of 2, 3 and 5; of 3, 4 and 6; of 3, 6 and 8.

6. Name the least common multiple of 2, 4 and 6; of 3, 5 and 6 ; of 1, 3 and 7.

7. Name the least common multiple of 3, 4 and 5; of 2, 5 and 6 ; of 4, 5 and 8.

8. Name the least common multiple of 5, 6 and 8; of 3, 7 and 8; of 5, 6 and 7.

WRITTEN EXERCISES.

ART. 101.-1. Find the least common multiple of 14, 24 and 36. Process.

Analysis.—Since 2 is a prime

factor of each of the numbers, 2)14, 24, 36

it is also a factor of their least 2) 7, 12, 18

common multiple. After divid3) 7, 6, 9

ing, there remain as the other 1, 2, 3

factors of the numbers 7, 12 2 x 2 x 3 x x 2 3= 504 and 18. 2 is a prime factor of

12 and 18, and is therefore a prime factor of their least common multiple. Dividing by 2, there remain 7, 6 and 9. 3 is a prime factor of 6 and 9, and hence is another factor of their least common multiple. Dividing by 3, there remain 7, 2 and 3, which are prime to each other. Hence, the product of the factors 2, 2, 3, 7, 2 and 3, is the least common multiple required. Proof.—504 - 14 36 ; 504 • 24 = 21; 504 • 36 = 14.

ART. 102.—Rule for finding the Least Common Multiple.Write the numbers in a horizontal line. Divide the given numbers by any prime number that is an exact divisor of two or more of them, and write the quotients and undivided numbers in a line beneath.

Continue the division until the quotients and undivided numbers are prime to each other. The product of the divisors and the numbers in the last horizontal line is the least common multiple required.

NOTE.-When one of the numbers is an exact divisor of one of the others, it may be disregarded in finding the least common multiple: Find the least common multiple of : 2. 26, 32 and 36.

5. 236, 284 and 320. 3. 42, 48 and 54.

6. 482, 520 and 564. 4. 120, 180 and 240. 7. 548. 624 and 680.

8. 148, 164, 248. 9. 308, 416 and 456. 10. 432, 528 and 564.

11. 18, 24, 48 and 56.
12. 32, 48, 60 and 72.
13. 28, 72, 84 and 96.

14. What is the least debt that can be paid with either 3 cent pieces, 5 cent pieces, 10 cent pieces, 25 cent pieces, or 50 cent pieces ?

15. Find the least sum with which I can buy sheep at $5, cows at $36, or horses at $120 each.

16. What is the least number of bushels of potatoes that can be divided into heaps of 12, 18

or 20 bushels ?

17. What is the shortest piece of rope that can be cut exactly into pieces either 15, 20 or 25 feet long?

18. What is the least number of acres in a farm that can be exactly divided into lots of 12, 16, 18 or 20 acres each ?

19. What is the least number of gallons that can exactly be measured by either of four casks holding 12, 15, 25 and 45 gallons respectively?

20. What is the least number of oranges that a teacher can exactly divide among either of four classes of children containing 16, 18, 20 and 24 respectively?

21. What is the least sum of money that will exactly pay either for cows at $48, oxen at $64, or horses at $120 each respectively ?

22. Five men start from the same point to walk around an island : the first can make the circuit in 3, the second in 4, the third in 6, the fourth in 8, and the fifth in 10 hours: in what time will they meet at the starting point ?

23. Illustrate by an original problem the method of finding the least common multiple.

Cancellation.

ART. 103.-1. How often is 2 times 4 contained in 4 times 4? 4 times 5 in 8 times 5 ? 4 times any number in 12 times that number?

2. How often is 3 times y contained in 9 times y? 5 times 6 in 15 times 6 ? 5 times any number in 20 times that number?

3. How often is ny times 8 contained in 14 times ? 8 times 10 in 24 times 10 ? 8 times any number in 40 times that number ?

4. What is the quotient of (10 x 12) = (5 x 12) ? Of (36 x 12) - (12 x 12) ?

5. What is the quotient of (27 x 30) = (9 x 30) ? Of (63 x 35) = (9 x 35) ?

6. What is the quotient of (144 x 120) - (12 x 120) ? Of (160 x 77) (40 x 77) ?

7. What is the quotient of (200 x 129) = (20 x 129) ? Of (360 x 137) = (36 x 137) ?

In finding the quotient, what numbers may be omitted from the dividend and divisor ?

Why does such omission not affect the value of the quotient ? Art. 72, prin. 4

ART. 104.—Cancellation is the process of shortening arithmetical operations by striking out equal factors from the dividend and divisor.

ART. 105.--Principle.-Dividing both dividend and divisor by the same number does not affect the quotient.

WRITTEN EXERCISES.

ART. 106.—1. Divide 4 x 5 x 12 by 2 x 3 x 3.

Process.

Analysis. - The divisor is

written under the dividend with 2 2

a line between. A x 5 x 12

2 = 63 Since the factor 2 in the di2 x 3 x 6

visor is a factor of 4 in the dividend, we strike it from both, leaving 2 in the dividend. Since the factor 6 in the divisor is a factor of 12 in the dividend, we strike it from both, leaving 2 in the dividend. The product of the remaining factors in the dividend is 20, and the divisor is 3. Hence, the quotient is = 63.

ART. 107.-Rule for Cancellation.-Cancel from the dividend and the divisor all common factors, and divide the product of the remaining factors in the dividend by the product of the remaining factors in the divisor.

NOTE.-Should all the factors in the dividend and divisor be canceled, the quotient is 1. If the 1 appears in the dividend it must be retained ; if in the divisor, it may be disregarded.

What is the quotient of:

2. 4 x 8 x 12 : 3 x 4 x 6 ? 3. 6 x 5 x 10 ; 4 x 3 x 6 ? 4. Y x 8 x 12 : 14 x 2 x 4 ? 5. 6 x 5 x 10 : 3 x 2 x 5 ? 6. 8 x 9 x 12 : 4 x 6 x 8? 7. 5 x 6 x 9 : 10 x 2 x 3 ? 8. 11 x x 12 ; 22 x 14 x 2 ? 9. 13 x 8 x 10 = 39 x 2 x 5 ? 10. 15 x 9 x 14 : 3 x 6 x 21 ? 11. How many dozen eggs worth 24 cents a dozen,

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