44. Applying the tests given above, find the prime factors

of 6480.

5)6480

6)1296 6)216 6)36 6

6 is used as a divisor to save time. The prime factors of 6 are 2 and 3. Therefore, 2 × 3 × 2 × 3 × 2 × 3 × 2 × 3 × 5 are the prime factors.

45. If the factors are alike, as 3 × 3 = 9, 3 x3 may be written 32, and read "3 square." The 2 which is written to the right and a little above the number is called the exponent, and indicates the power of the number, or the number of times it is to be taken as a factor. 33 = 3 × 3 × 3, and is read "3 cube." 34 is read "3 to the fourth power."

NOTE. No sign has been used between a and x in ax, but multiplication is understood.

When no sign is expressed between two letters or between a number and a letter, multiplication is understood; e.g., 5 a means 5 times a.

18. What does 5 (a - b) mean?

46. The sign √ (called radical sign) indicates that one of the equal factors of a number is to be found, or in other words, that the square root is to be taken; e.g.,

the square root of 9, or 3.

means

NOTE. For the method of extracting the square root see Appendix.

47. A common divisor of two numbers is a number that will exactly divide each of them.

48. The greatest common divisor (G. C. D.) of two or more numbers is the largest number which will exactly divide each of them.

49. To find the G. C.D. of two numbers, divide the larger by the smaller, the first divisor by the remainder, and so on until there is no remainder. The last divisor is the G.C.D. Find the G. C. D. of 576 and 198.

198)576(2 396

180)198(1
180

18)180(10
180

Divide the larger number by the smaller. The quotient is 2, with a remainder of 180. Divide this remainder into the first divisor, 198. The result gives a remainder of 18 which goes exactly into 180. The G. C. D. is the last divisor, 18.

If the numbers are not too large, the process may be shortened as in the following example.

Find the G. C. D. of 66 and 42.

Do not indicate all the steps in the division, but merely the remainders. The last divisor, 6, goes into 18 exactly, hence 6 is the G. C. D.

6

The G. C. D. can often be found by inspection. It must be a factor of the difference between the two numbers; e.g., find the G. C. D. of 24 and 45. The difference between them is 21, whose factors are 3 and 7. Since 7 is not a divisor of both numbers, 3 is the G. C. D.

Find (by inspection whenever possible) the G. C. D. of :

8. 144, 111

11. In a sawmill

there are three tree trunks of lengths What are the longest uniform boards that can be cut from them without waste?

32, 48, and 80 ft.

12. A field is 160 rods long and 80 rods wide. Boards of what length under 15 ft. can be used to fence it without waste? (1 rod = 161 ft.)

50. A common multiple of two or more numbers is a number that can be divided by each of them without a remainder; thus, 24 is a common multiple of 2, 3, and 4.

51. The least common multiple (L. C. M.) of two or more numbers is the smallest number that can be divided by each of them without a remainder; thus, 20 is the L. C. M. of 2, 4, and 5.

Find the L. C. M. of 14, 49, 77, and 22.

52. Cancellation is useful when the product of several numbers is to be divided by the product of other numbers. Divide the product of 15 × 9 by the product of 5 × 3 × 2.

3

15 x 9. 5×3×2

9

2'

Divide any number on one side of the line by any number on the other side of the line. or 41 When unable to divide further, multiply the numerators together for a new numerator and the denominators for a new denominator.

NOTES. 1. When no uncanceled number on one side of the line will divide an uncanceled number on the other side of the line, divide by any factor common to both.

2. Make use of cancellation whenever possible. Do not multiply or divide numbers until you are sure no cancellation is possible.

Use cancellation in the following problems:

1. Divide 81 x 25 × 34 × 30 by 21 × 5 × 6 × 17. 2. Divide 39 × 91 × 96 by 114 x 95 x 160.

3. Divide 64 × 76 × 45 × 68 by 114 × 160 × 30.

4. A large sheet of paper cuts into 12 small sheets. How many packages of 480 small sheets can be cut from 72,600 large sheets?

5. A plot of ground 320 rods by 400 rods is laid off in lots 50 ft. x 160 ft. How many such lots can be laid off?

6. 12 cases of books, 140 books to the case, are to be repacked in cases holding only 84 books each.

cases will be required?

How many

7. If 24 tons of coal cost $135.75, how much will 60 tons cost at the same rate?

FRACTIONS

53. Fractions are of two kinds, common and decimal.

54. The difference between common and decimal fractions is that a common fraction may have any number for its denominator, while the denominator of a decimal fraction is always 10, or some power of 10, as 100, 1000, etc.

55. The denominator of a decimal is not written, but is indicated by the number of places to the right of the decimal point; e.g., .125 is read one hundred twenty-five thousandths (125). 1000 is the denominator because there are three places to the right of the decimal point.

1000.

56. (a) Common and decimal fractions are treated together, because speed and accuracy in handling fractions come from the ability to interchange them rapidly.

(b) In the work which follows, the term "fraction" is used to indicate common fractions.