IV. If more than two numbers are given, first find the greatest common divisor of two of them, and then of this divisor and one of the remaining numbers, and so on to the last; the last common divisor found is the greatest common divisor of all the given numbers.

10. What is the greatest number that will divide 3281 and 10778? 10353 and 14877 ?

11. What is the greatest number that will divide 620, 1116, and 1488? 396, 5184, and 6914?

12. A man having a piece of land, the sides of which are 240 feet, 648 feet, and 420 feet, wishes to inclose it with a fence having panels of the greatest possible uniform length; what will be the length of each panel?

13. A farmer wishes to put 231 bushels of corn, 393 bushels of wheat, and 609 bushels of oats into the largest bags of equal size, that will exactly hold each kind. How many bushels must each bag hold?

14. A forwarding merchant has 15292 bushels of wheat, 1520 bushels of corn, and 504 bushels of beans, which he wishes to ship, in the fewest bags of equal size that will exactly hold either kind of grain; how many bags will it take?

15. Three persons have respectively $630, $1134, and $1386, with which they agree to purchase horses, at the highest price per head, that will allow each man to invest money. How many horses can each man buy?

all his

16. How many rails will inclose a field 5850 feet long by 1729 feet wide, the fence being straight, and 7 rails high, and the rails of equal length, and the longest that can be used?

172. 1. What numbers between 5 and 30 are exactly divisible by 4? By 6? 7? 8? 9?

2. What numbers less than 40 are exactly divisible by 7? 3. What prime factors are common to 6, and 5 times 6? 4. Name some numbers exactly divisible by 4 and 6; by 3 and 7; by 5 and 7; by 8 and 10.

5. By what three prime numbers can 42 be divided? 6. Name some numbers of which 3 and 4 are factors. 7. Find the least number exactly divisible by 3, 4, and 5.

DEFINITIONS AND PRINCIPLES.

173. A Multiple of a number is a number exactly divisible by the given number; or, it is any product or dividend of which a given number is a factor.

1. A number may have an unlimited number of multiples. 2. A number is a divisor of all its multiples, and a multiple of all Its divisors.

174. A Common Multiple of two or more given numbers is a number exactly divisible by each of them.

175. The Least Common Multiple of two or more given numbers is the least number exactly divisible by each of them.

Two or more numbers can have but one least common multiple. 176. PRINCIPLES.-1. A multiple of a number contains each of the prime factors of that number.

2. A common multiple of two or more numbers contains each of the prime factors of those numbers. Hence,

3. The least common multiple of two or more numbers is the least number that contains each of the prime factors of those numbers.

4. A common multiple of two or more numbers may be found by multiplying the given numbers together.

WRITTEN

EXERCISES.

177. To find the least common multiple.

FIRST METHOD.

1. Find the least common multiple of 30, 42, and 66.

OPERATION.

30= 2 × 3 × 5
42 = 2 × 3 × 7
66=
2 x 3 x 11

2×3×11x7x5=2310

ANALYSIS.-The least common multiple cannot be less than the largest number 66, since it must contain 66; hence it must contain all the prime factors of 66, which are 2, 3, and 11. (PRIN. 1.) But the least common multiple of

66 must also contain all the prime factors of each of the other numbers, and since the prime factors 2 and 3 of 66 are common also to 42 and 30 omit them, and annex the factors 7 and 5 to those of 66, and the series 2, 3, 11, 7, and 5 are all the prime factors of the given numbers, and their product 2×3× 11 × 7×5=2310, is the least common multiple of the given numbers. (PRIN. 3.)

2. Find the least common multiple of 24, 42, and 17. 3. Find the least common multiple of 8, 12, 20, and 30. 4. Find the least common multiple of 10, 45, 75, and 90. RULE.-I. Resolve each of the given numbers into its prime factors.

II. Multiply together all the prime factors of the largest number, and such prime factors of the other numbers as are not found in the largest number, and their product will be the least common multiple.

Find the least common multiple

5. Of 30, 66, 78, and 42.

7. Of 16, 60, 140, and 210.

6. Of 21, 30, 44, and 126. I 8. Of 16, 48, 80, 32, and 66.

OPERATION.

218 24 54

SECOND METHOD.

178. 1. Find the least common multiple of 18, 24, and 54. ANALYSIS.-Write the numbers in a hori zontal line, with a vertical line at the left. Since 2 is a prime factor of one or more of the given numbers, it must also be a factor of the least common multiple of those numbers. (PRIN. 3.) Hence, divide by 2 and write the quotients underneath. For a like reason divide again successively by 3 and 3, writing the quotients and undivided numbers in a line below, omitting to write any quotient when it is 1.

3 9

12

27

3

3

4

9

4

3

Since there is no factor common to 4 and 3, they are prime to each other, and hence the divisors 2, 3, and 3, with the numbers 4 and 3 in the last line, are all the prime factors of the given numbers, and their product 216 is the least common multiple. (PRIN. 3.)

If in any example, any of the smaller numbers are exactly contained in the larger, they may be omitted in finding the least common multiple, inasmuch as a number that will contain a given number, will contain any factor of that number.

Thus, if required to find the least common multiple of 8, 12, 24, 72, and 120, omit all the numbers except 72 and 120, since the others are factors of these, and the least common multiple of 72 and 120, will be the least common multiple of all the numbers.

2. Find the least common multiple of 32, 34, and 36. 3. Find the least common multiple of 84, 100, and 224. RULE.-I. Write the numbers in a horizontal line, omitting such of the smaller numbers as are factors of the larger, and draw a vertical line at the left.

II. Divide by any prime factor that will exactly divide two or more of the given numbers, and write the quotients and undivided numbers in a line underneath.

III. In like manner divide the quotients and undivided numbers until they are prime to each other.

IV. The product of the divisors and the final quotients and undivided numbers, is the least common multiple,

What is the least common multiple

4. Of 4662, and 5698? N6. Of 24, 10, 32, 45 and 25? 5. Of 312, 260, and 390? 17. Of 153,204,102, and 1020? 8. Find the least common multiple of the first eight even numbers.

9. Find the least common multiple of the first five odd numbers.

10. What is the least number of oranges that can be equally distributed among 16, 20, 24, or 30 boys ?

11. What is the shortest piece of rope that can be cut exactly into pieces either 15, 18, or 20 feet long?

12. What is the smallest sum of money which can be exactly expended for books at $5, or $3, or $4, or $6 each ?

13. What is the product of the least common multiple of 12, 16, 24, and 32, multiplied by their greatest common divisor?

14. Divide the least common multiple of 7, 42, 6, 9, 10, and 630, by the greatest common divisor of 110, 140, and 680.

-15. What is the smallest sum of money which can be exactly expended for sheep at $8, or cows at $28, or oxen at $54, or horses at $162 each ?

16. What is the smallest quantity of grain that will fill an exact number of bins, whether they hold 36, 48, 80, or 144 bushels ?