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a day, and B 35 miles. In how many days will they be 84 miles apart. How soon will they be 96 miles apart?

35. If a mechanic earns $775 a year, and pays $240 for rent, and 390 for family expenses, how many years will it take him to save $2900?

36. If a mechanic earns $550 a year, and his expenses are $425, how many years will it take him to buy a farm of 50 acres at $75 an acre?

37. A dairy-man in packing 765 pounds of butter found that it filled 18 tubs, with 9 pounds left over.

did he put in each tub?

How much

38. If three steers weigh respectively 875 pounds, 944 pounds, and 1025 pounds, what is their average weight?

39. If a merchant's sales amount to $25225 in a week of 6 days, what are his average daily sales?

40. A farmer bought 25 acres at $80 an acre, and 30 acres at $75 an acre? What was the average price per acre?

41. How many casks, each holding 21 gallons, can be filled from 25 hogsheads of wine, containing 84 gallons each?

SECTION VII.

Properties of Numbers.

1. What is the product of the factors 3 and 3? 3 and 5?

2. The factors are 3 and 4. What is the product?

3. How many are 2 times 1? 3 times 1? 5 times 1? 4. How many ones in 3? In 5? In 7?

5. What numbers multiplied together produce 3? 5? 7?

6. What two numbers other than itself and 1 multiplied together produce 4? 6? 8? 10?

7. What two numbers other than itself and 1 divide 4 without a remainder? 6? 8? 10? 18? 24?

8. Of what number are 2 and 3 the factors? 2, 2, and 3? 2, 3, and 5? 2, 2, 3, and 5?

9. Name the smallest numbers other than 1 that will divide 6 without a remainder; 12; 30; 60.

10. Which are all the numbers that divide 9 without a remainder? 12? 18? 20? 24?

11. Name five numbers each of which is produced only by multiplying together itself and 1.

12. Name five numbers each of which is produced by multiplying together other numbers than itself and 1.

13. Name the numbers from 0 to 20 that can be divided by 2 without a remainder.

14. Name all the numbers from 0 to 20 that cannot be divided by 2 without a remainder.

Definitions.

163. The Factors of a number are those numbers which, multiplied together, produce that number.*

Thus, 2 and 3 are factors of 6; for, being multiplied together they produce 6; so, also, 2, 3, 5 are the factors of 30.

Note.-The terms number, factor, divisor, and multiple in this Seetion are used in the sense of integers.

164. A Prime Number is a number that has no other factor than 1 and itself.

Thus, 5 is a prime number; so, also, are 7, 11, 13, 17, etc.

165. A Composite Number is a number that has other factors than 1 and itself.

*Sections numbered 1-162, inclusive, will be found in Part I. of the Model Elementary Arithmetic.

Thus, 15 is a composite number, since 15 205 X 4, or 10 X 2, etc.

3 X 5; and 20, since

166. An Even Number is a number that can be divided by 2 without a remainder.

All even numbers end with 0, 2, 4, 6, or 8.

167. An Odd Number is a number that cannot be divided by 2 without a remainder.

All odd numbers end with 1, 3, 5, 7, or 9.

Factors, or Divisors.

1. Name all the prime numbers from 0 to 20.

2. Name all the composite numbers from 0 to 20. 3. What is the product of the prime numbers 2 and 2? 2 and 3? 3 and 5? 2, 2, and 3? 2, 3, and 5?

4. Of what prime numbers is 4 the product? 6? 15? 12? 30? 16? 33? 18? 36?

5. What prime numbers are factors of 18? 24? 36? 6. Name the smallest exact divisors of 27; 32; and 42. 7. Name all the smallest prime numbers whose continued product is 36.

8. Name all the smallest prime numbers that will exactly divide 36.

9. Name all the numbers which, multiplied together, produce 12; 36; 48.

10. Name all the numbers that will exactly divide 12; 36; 48.

11. Name two composite numbers that are factors of 48. Name the smallest factors of these two composite numbers. 12. Name all the prime factors of 48. Name every factor of 48.

Definitions.

168. An Exact Divisor is a number that divides any given number without a remainder.

Thus, 5 is an exact divisor of 25; and 8 of 56.

1. The exact divisors of a number are also the factors of that number. 2. The terms Factor and Divisor differ only in use; factor suggesting the process of multiplication, and divisor the process of division. Note. A number exactly divides another when it is contained

in that other without a remainder.

Thus, 5 exactly divides 10, 15, 30, 75, etc.

A number can be exactly divided by another when it contains that other without a remainder.

Thus, 24 can be exactly divided by 2, 3, 4, 6, 8, and 12.

169. A number can be exactly divided by 2 when it ends with 0, 2, 4, 6, or 8.

Thus, 10, 22, 34, 456, 678 can be exactly divided by 2 (166).

170. A number can be exactly divided by 5 when it ends with 5 or 0.

Thus, 15, 70, 345, 670 can be exactly divided by 5.

171. A number can be exactly divided by 3 when the sum of the units expressed by its figures can be divided by 3. Thus, 3579 can be exactly divided by 3, since the sum of 3, 5, 7, 9, or 3+5+7+ 9 = 24, can be divided by 3.

172. A Prime Factor is a factor that is a prime number (164).

Thus, 5 is a prime factor of 15; and 7 of 42.

173. A Composite Factor is a factor that is a composite number (165).

Thus, 6 is a composite factor of 12; and 14 of 42.

174. Factoring is the process of separating a composite number into its prime factors.

Since 1 is a factor of every number, it is not regarded either in naming or in finding the prime factors of numbers.

175. Principles.

I. Every composite number is the product of all its prime factors.

II. Every factor of a number is an exact divisor of that number.

III. Every number can be exactly divided only by its prime factors, and by the product of any two or more of them.

Written Exercises.

176. Example 1. What are the prime factors, or divisors, of 456?

SOLUTION.

2)456

2)228

2)114

3)57

19

EXPLANATION.-Since the given number is an even number, it can be divided by the prime factor 2 (169).

For the same reason, divide the quotient, 228, by 2; and the next quotient, 114, by 2.

Since the sum of 5 and 7 can be exactly divided by 3, divide 57 by 3 (171), giving the quotient 19, which is a prime number, and cannot be separated into factors (164).

PROOF.

2 X2 X2 X3 X 19 = 456

Hence, the divisors 2, 2, 2, 3, and 19 are all the prime factors of 456.

177. Example 2.—Find all the factors, or divisors, of 90.

SOLUTION.

2)90

2, 3, 3, 5

2 X 3

6

3)45

2

= 10

[blocks in formation]

EXPLANATION. - Since the given number is an even number, etc.

Hence, 2, 3, 3, and 5 are the prime factors of 90.

Since the only composite factors of 90 are the products of any two or more of its prime factors, 6, or 2 times 3; 10, or 2 times 5; 9, or 3 times 3; 15, or 3 times 5; 18, or 2 times 3 times 3; and 45, or 3 times 3 times 5, are the composite factors of 90.

Hence, 2, 3, 3, 5, 6, 9, 10, 15, 18, and 45 are all the factors, or divisors, of 90 (175, III.).

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