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8. 21.4 x 3.2= 11. 69.12:16= 14. 235.62 46.3= 9. 23.6 x 4.3=

12. 73.78:17= 15. 223.155_9.5= 10. 16.9 x 7= 13. 210.56 : 28= 16. 917.28 : 7.2=

17. How many square yards in the four walls of a room 18 feet long, 15 feet wide, and 9 feet high?

18. Change 2880 rods to miles. To yards. To feet.

19. Find the number of days from Oct. 16, 1899, to April 1, 1900.

20. Find the number of sods 18 inches by 10 inches that will be needed to cover a lawn 12 feet long and 10 feet wide.

Find the cost of:

21. 36,400 lb. of hay at $14.50 per ton. 22. 9500 shingles at $3.75 per M. 23. 3460 cabbages at $3 per C. 24. 3846 lb. of coal at $3 per ton (2000 lb.). 25. 4620 ft. of boards at $36 per M. 26. 4860 lb. of wheat at 75¢ per bushel (60 lb.). 27. If į of an acre of land cost $12, find the cost of 142

acres.

28. Find the cost of 4 of 28 ounces of butter at 32 ¢ a pound.

29. At 33; ¢ a pair, how many pairs of cuffs can be bought for $8?

30. How much will 94 yards of lace cost at 36¢ a yard ? 31. Find the cost of 186 eggs at 20¢ a dozen.

FACTORS, DIVISORS, AND MULTIPLES

PRIME AND COMPOSITE NUMBERS

LESSON 21

1. What two numbers when multiplied together will make 6? 8? 9? 10 ? 12? 14? 15 ?

2. What three numbers when multiplied together will produce 8? 12? 16 ? 18? 20 ? 27 ? 28 ? 30 ?

All numbers that can be formed by multiplying together two or more integral numbers each greater than 1 are called Composite Numbers. The numbers that produce composite numbers are called Factors (makers). Thus, 3 and 4 are factors of 12, and 3 and 5 are factors of 15.

Numbers that cannot be produced by multiplying together two or more integral numbers, each greater than 1, are called Prime Numbers.

3. Name the composite numbers between 1 and 45. 4. Name the prime numbers from 1 to 45. 5. Is 16 a prime or a composite number? Why? 6. What two numbers multiplied together make 12 ? 21 ?

7. Are the two factors of 12 prime or composite numbers ? Are 3 and 7 (factors of 21) prime numbers ? Why? Factors that are prime numbers are called prime factors.

8. What composite number is formed from the two prime factors 2 and 3? 2 and 5 ? 3 and 5? 2 and 7 ? 2, 3, and 5 ? 2, 3, and 7 ? 2, 5, and 7 ?

Find the composite numbers that equal the following prime factors :

9. 2 x 3 x 5 = 12. 3x3 x 5= 15. 2x2x3 x 5= 10.2 x 2 x 5= 13. 3 x 5 x 5 = 16. 3 x 3 x 3 x 5= 11. 2 x 2 x 7= 14. 2 x 3 x 7= 17. 2 x 2 x 3 x 11=

FACTORING

LESSON 22

Factoring is the process of finding the factors, or makers, of a composite number. Name the two prime factors of the following: 1. 4= 2; 2

5. 15
9. 21

13. 55 =
2. 9= 3; 3 6. 25 =

10. 26

14. 33 = 3. 10:

7. 35
ll. 34

15. 77 =
4. 14 =

8. 22
12. 49

16. 57 =
Name the three prime factors:
17. 12 = 2; 2; 3

21. 44

25. 50 = 29. 68= 18. 20 =

22. 45
26. 52:

30. 78 19. 27 =

23. 63
27. 70

31. 42 20. 28=

24. 66
28. 75 =

32. 99

=

The prime factors of any composite number may be found by dividing the given number by any prime number that will exactly divide it, and the quotient by any prime number that will exactly divide it, and thus continuing to divide the resulting quotient until it becomes a prime number. The several divisors and the last quotient will be the required prime factors.

33. Find the prime factors of 210:

OPERATION I

OPERATION II

OPERATION III

2 | 210
7 210

5 | 210
3 195
5 30

3 42
5 35
3 6

2 14
7
2

7 The prime factors of 210, arranged according to their value, are 2, 3, 5, 7.

Find the prime factors of the following:
34. 40, 72, 84, 92, 95, 110, 160, 176, 360, 410, 876.

GREATEST COMMON DIVISOR

LESSON 23

1. Name a number that will exactly divide 10, 21, 25, 28, 32.

2. Name a number that will exactly divide both 9 and 12, 10 and 15, 12 and 16, 24 and 36.

A number that will exactly divide two or more numbers is called a Common Divisor of those numbers. Thus, 4 is a common divisor of 28 and 36, since it exactly divides each of them.

3. Name the largest number that will exactly divide 16 and 24, 32 and 40, 36 and 45.

The largest number that will exactly divide two or more numbers is called the Greatest Common Divisor of those numbers. Thus, 9 is the greatest common divisor of 18 and 27, since it is the largest number that will exactly divide each of them.

The letters G.C.D. are generally used to denote Greatest Common Divisor.

Name at sight the G. C.D. of:
4. 14, 16
7. 32, 44

10. 48, 64, 80
5. 27, 30
8. 72, 80

11. 54, 63, 81 6. 32, 40 9. 84, 96

12. 60, 72, 96 13. Name all the divisors of 12, 14, 16, 18, 32, 40.

14. Name all the divisors common to 12 and 18, 24 and 36, 60 and 72, 48 and 60, 32 and 40.

LESSON 24

When the numbers are not too large we may easily see, by inspection, the G.C. D. of two or more numbers by first arranging all the divisors of the given numbers in pairs as follows:

60

= 2 x 30, 3 x 20, 4 x 15, 5 x 12, 6 x 10

72 = 2 x 36, 3 x 24, 4 x 18, 6 x 12, 8 x 9 Arranging in order first the divisors of 60: 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and then the divisors of 72: 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, we readily see that 12 is the G.C. D. of 60 and 72.

The prime factors of 60 are: 2, 2, 3, 5.

The prime factors of 72 are: 2, 2, 2, 3, 3. It will be seen from the above work that the prime factors common to 60 and 72 are 2 twos and 1 three, the product of which, 12, is the G.C. D.

1. Find the G.C.D. of 48 and 56.

MODEL. 48 2 x 2 x 2 x 2 x 3

56 2 x 2 x 2 x 7

2 x 2 x 2 = 8, G. C. D. Find the G.C.D. of: 2. 40, 48 5. 40, 96

8. 45, 75 3. 72, 84 6. 56, 60

9. 48, 80 4. 64, 72 7. 96, 72

10. 35, 75

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