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8. A grain dealer has 2722 bushels of wheat, 1822 bushels of corn and 1226 bushels of beans which he wishes to “ship” in the smallest number of bags of equal size. Find the size of the bags.

9. Find the size of the largest equal packages that will contain without mixing 60 pounds of one kind of tea, 75 pounds of a second kind, and 100 pounds of a third kind.

10. There is a triangular field whose sides are 288, 450, and 390 feet. What is the least number of rails that will enclose it, with a fence 5 rails high ?

COMMON DIVIDENDS.

INDUCTIVE STEPS.

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1. In the expression, 30 = 5= 6, which number is the dividend ? Which the divisor?

2. In the expression, 30 - 6 = 5, which is the divisor?

3. Since 30 is divisible by both 5 and 6, it is called a common dividend of 5 and 6.

4. Can you find a number less than 30 that is a common dividend of 5 and 6?

5. Since such a number cannot be found, 30 is called the Least Common Dividend (L. C. Dd.) of 5 and 6.

6. What are the prime factors of 5 and 6? 2, 3, 5. What are the factors of 30 ? 2, 3, 5.

7. Hence we see that the L. C. Dd. of two or more numbers is composed only of the factors of those numbers.

8. The prime factors of 42 are 2, 3, 7. Is 42 the L. C. D. of 6 and 7? Why? Of 3 and 14? Why? Of 2 and 21 ? Why?

9. Does 42 contain any other factors than those of 6 and 7, 3 and 14, or 2 and 21 ?

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6 2 X 3
10. Numbers 8 2 X 2 X 2 prime factors.

20 = 2 X 2 X 5
What prime factor is common to all the numbers ?
What prime factor is common to 8 and 20 only?
Is this 2 a different 2 from the other?

What three prime factors are not common to any two of the numbers?

How many different prime factors are common?
How many are not common?
How many different factors in all ?
Name them.
Which one is common to all the numbers ?
Which one is common to two of the numbers ?
Which three are not common ?

11. Three classes of different prime factors are to be recoghized : (1) Factors that are common to all the numbers; (2) Factors that are common to some of the numbers ; (3) Factors that are not common to some of the numbers.

12. Can you form a L. C. Dd. of two or more numbers without using all the different prime factors of those numbers ? Why not?

A factor common to all the numbers will be taken how often as a factor of the L. C. Dd.?

A factor common to some of the numbers will be taken how often as a factor of the L. C. Dd. ?

Will a factor not common to some of the numbers be used as a factor of the L. C. Dd.?

DEFINITIONS.

1. A Dividend of a number exactly contains that number.

NotE.—The word Multiple has commonly been used instead of Dividend.

2. A Common dividend of two or more numbers exactly contains each of them.

3. The Least common dividend (L. C. Dd.) of two or more numbers is the least number that exactly contains each of them.

PRINCIPLE. The L. C. Dd. of two or more numbers equals the product of all the different prime factors of the numbers, and no other factors.

First Method. What is the L. C. Dd. of 20, 30, 70 ? Process.

Explanation. 20 = 2 X 2 X 5

We first resolve the numbers into their prime

factors. The L. C. Dd. equals the product of 2 X 3 X 5

all their different prime factors. The factors 70 = 2 X 5 X 7 common to all the numbers are 2 and 5. The

factors not common to some of the numbers L. C. Dd. =- 2 X

are 2. 3, and 7. Hence the factors of the L. 5 X 7 X3 X 2.

C. Dd are 2, 5, 2, 3, and 7, and the L. C. Dd.

- 2 X 2 X 3 X 5 X 7= 420.

30

RULE.
Resolve the given numbers into their prime factors.

Select all the different factors, common and not common, and find their product.

EXERCISES. Find the L. C. Dd. of: 1. 28, 21, 14.

10. 30, 32, 36. 2. 60, 48, 36.

11. 56, 72, 96. 3. 28, 32, 64.

12. 20, 24, 36. 4. 16, 24, 36.

13. 22, 33, 55. 5. 15, 45, 60.

14. 36, 40, 48. 6. 45, 30, 72.

15. 30, 50, 80. 7. 20, 24, 27.

16. 25, 45, 75. 8. 35, 40, 42.

17. 36, 48, 64. 9. 80, 60, 200.

18. 100, 450, 900.

Since a common factor enters but once into the L. C. Dd., an abridged method is usually adopted.

Second Method.

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Process.

Explanation. 2 8, 6, 20

2, a common factor of 8, 6, and 20, is a 2 4, 3, 10

factor of the L. C. Dd. 2, a common factor

of 4 and 10, is a factor of the L. C. Dd. 2, 3, 2, 3, 5

and 5, in the last line, are the factors that are

not common. L. C. Dd. = 2 X

Hence L. C. Dd. = 2 X 2 X

2 X 3 X 5 = 120. 2 X 2 X 3 X 5.

Find the L. C. Dd. of the following: 1. 15, 60, 75.

9. 12, 26, 52. 2. 8, 12, 40.

10. 18, 24, 36. 3. 14, 35, 56.

11. 12, 18, 21. 4. 9, 18, 27, 54.

12. 18, 36, 72, 108. 5. 27, 36, 45, 90.

13. 12, 48, 36, 70. 6. 18, 21, 24, 27.

14. 17, 51, 34, 85. 7. $240, $270, $180, $150. 15. 21, 24, 26, 28, 30. 8. 9, 10, 14, 15, 18.

16. 45, 50, 60, 63, 84.

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Third Method.

The two foregoing methods of finding the L. C. Dd. show tiat two kinds of factors come into play,-factors common and factors not common.

Since the G. C. D. of two numbers is the product of their common factors, the quotients of the numbers divided by the G. C. D. are either the prime factors not common or the

products of those factors. Hence, to find the L. C. Dd. of two numbers not readily factored :

1. Find the G. C. D. of the numbers. 2. Divide the numbers by the G. C. D. 3. Find the product of the G. C. D. and the quotients.

1. Find the L. C. Dd. of 849 and 1132.

Process. (1.)

(2.) 849) 1132 (1

283 ) 849, 1132 849

3, 4 G. C. D.= 283 ) 849 (3

849

(3.)
L. C. Dd. 283 X 3 X 4 = 3396.

2. How many kinds of factors does the L. C. Dd. of 849 and 1132 contain ?

3. Why did we first find the G. C. D. of 849 and 1132 ? 4. Why did we divide 849 and 1132 by 283 ?

Note. — When the L. C. Dd. of more than two numbers is required, first find the L. C. Dd. of two of them, and then the L. C. Dd. of the result and a third number, and so on.

5. Find the L. C. Dd. of:

1. 261 and 319. 15. 9797 and 10,403.
2. 731 and 817. 16. 9523 and 11,663.
3. 527 and 589. 17. 2479 and 3589.
4. 91 and 117.

18. 3045 and 6195.
5. 135 and 144. 19. 568 and 712.
6. 169 and 221. 20. 11,023 and 6493.
7. 357 and 612. 21. 1485 and 2160.
8. 1417 and 1469. 22. 30,072 and 133,784.
9. 17,640 and 18,375. 23. 9,144,407 and 10,347,059.
10. 1110 and 777. 24. 1177, 1391, 1819.
11. 4087 and 4757. 25. 2943, 2616, 4578.
12. 1728 and 1898. 26. 31, 124, 217, 310.
13. 321 and 314,259. 27. 113, 452, 1000, 1492.
14. 7854 and 86,394. 28. 135, 144, 356, 612.

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