165. When the same number is used several times as a factor of a number, the number of times it is so used is indicated by a small figure called an exponent.

=

5 × 5. 52 is read "Five to the second power"; 25: =

Thus, 52 2 × 2 × 2 × 2 × 2.

25 is read "Two to the fifth power."

DIVISIBILITY OF NUMBERS.

166. Numbers divisible by 2 are even numbers; all other numbers are odd numbers. All prime numbers except 2 are odd.

The factors of a number may be found by division. The divisibility of numbers by some factors can be determined by the following tests: 1. A number is divisible by 2 if the units are divisible by 2.

=

2. A number is divisible by 3 if the sum of its digits* is divisible by 3. Thus, 285 is divisible by 3, for 2 + 8 +5: 15 is divisible by 3. 3. A number is divisible by 4 if its tens and units together are divisible by 4. Thus, 6724 is divisible by 4, while 6731 is not.

4. A number is divisible by 5 if the units' figure is either 0 or 5. 5. A number is divisible by 6 if it is divisible by 2 and by 3.

6. A number is divisible by & if its hundreds, tens, and units together are divisible by 8. Thus, 6728 is divisible by 8, while 6724 is not.

7. A number is divisible by 9 if the sum of its digits is divisible by 9.

8. A number is divisible by 11 if the sums of its alternate digits are equal, or if their difference is divisible by 11. Thus, 1782 and 1859 are divisible by 11, while 4987 is not.

9. A number is divisible by a composite number, if it is divisible by each of the factors of the composite number. Thus, 3825 is divisible by 15, for it is divisible by 3 and by 5.

NOTE.Reasons for these tests are given in the Supplement, Art. 6.

167. Exercises for Written Work.

1. Turn to page 89 and, using the tests above, write the numbers expressed in w, lines a, b, c, etc., that contain as a factor, 2; 3; 4.

* A digit here means the number denoted by a figure independently of its place or order.

2. Write the numbers expressed in x, lines a, b, c, etc., that contain as a factor, 5; 6; 8.

3. Write the numbers expressed in y, lines a, b, c, etc., that contain as a factor 9; 10; 11.

Find by inspection, or by trial when necessary, all the factors not greater than 12 that are contained in the numbers given below:

4. 25,785 7. 39,490 5. 38,948 8. 25,404 6. 27,645 9. 82,936

10. 90,273 13. 80,661 11. 50,809 14. 22,891

12. 39,872 15. 56,784

168. To find the prime factors of a number.

Illustrative Examples. (1) What are the prime factors of 735 ? (2) What are the prime factors of 409?

(1) Applying the tests (Art. 166) to the given number, we find that 2 is not a factor of 735, but that 3 is, and by dividing, we see that 7353 x 245.

Continuing this process, we find that 245-5 × 49, and that 49=7x7. Therefore 735 = 3 × 5 × 7 × 7, and the prime factors are 3, 5, 7, and 7.

(2) Applying the tests (Art. 166) to the second example, we find that 409 is not divisible by 2, 3, or 5. We then try to divide by the other prime numbers in order until we reach 23, when we see that the quotient is less than the divisor. There can then be no prime factor in 409 greater than 23, for if there were, there would be another factor (the quotient) less than 23, which we should have found before reaching 23. The number 409 is, therefore, prime.

As 735 is found to equal the product of all its prime factors, so will it always be found that a composite number equals the product of all its prime factors.

169. From the preceding examples may be derived the following

Rule.

To separate a number into its prime factors:

1. Divide the given number by one of its prime factors.

2. Divide the quotient thus obtained by one of its prime factors; and so continue dividing until a quotient is obtained that is a prime number.

3. This quotient and the several divisors are the prime factors sought.

Proof.

Multiply together the prime factors thus found. The prod uct should equal the given number.

NOTE. If no prime factor is readily found by which to divide, try the several prime numbers in order. If no prime factor is found before the quotient becomes less than the trial divisor, the given number is prime.

Examples for Written Work.

170. Separate into prime factors the following numbers:

Select the prime numbers and find the prime factors of

the composite numbers among the following:

What is a factor of a number? What is a composite number? a prime number? a prime factor?

What is an even number? an odd number? What numbers are divisible by 2? 3? 4? 5? 6? 8? 9? 10? 11?

How can you find the prime factors of a number? A composite number equals what product?

CANCELLATION.

171. Illustrative Examples.

(1) Multiply 5 by 3 and divide the product by 3. (2) Divide the product of 5 and

(1) If any given number is multiplied by a number and the product divided by the same number, the two operations offset, or cancel, each other, and the result is the given number. Hence, both operations may be omitted. Ans. 5

(2) 6 2 x 3; therefore 2 is a factor of both the dividend and divisor, and may be struck out or canceled, and 5 may be multiplied by 3 only. Ans. 15.

172. Cancellation is the process of striking out equal factors in dividend and divisor.

NOTE. — The pupil should shorten his work by canceling, whenever it is possible.

58. If a bar of iron 8 feet long weighs 36 pounds, what will a bar of iron of the same thickness, and 100 feet long, weigh?

59. If a tree 69 feet high casts a shadow 90 feet, what length of shadow will a tree 92 feet high cast at the same time of day?

60. If the work of 6 men is equal to the work of 9 boys, how many men's work will equal the work of 21 boys?

61. A yachting party has provisions for 12 persons 60 days. How long will it last 21 persons?

62. A merchant exchanged 126 pounds of sugar at 6 cents a pound for eggs at 28 cents a dozen. How many dozen did he receive?

63. How many yards of cloth 24 inches wide will be required to line six yards of cloth 28 inches wide?

64. A rectangular plat of ground 63 feet by 48 feet is turfed with sods 6 feet long and 1 foot wide.

sods does it take?

How many

65. 100 rolls of dimes, 20 to a roll, were given in exchange for a roll of $5 bills. How many bills were there in the roll?

66. How many shares of stock worth $175 per share will pay for 750 shares at $77 per share?

67. The freight on 19,000 pounds of coal was $7.60. What is the freight on 100 long tons at the same rate?

68. How many years will be required to pay a debt of $3,900 at the rate of $650 in 5 years?

69. I sold 20 barrels of apples at $2.50 a barrel and spent the money thus obtained for cloth at $0.50 a yard. This I afterwards sold at 60 cents a yard and bought a horse with the proceeds. How much did I pay for the horse?

70. On a certain railroad, a passenger counted 25 telegraph poles passed in 3 minutes. If the poles were set 88 yards apart, how many miles per hour was the train going?