28. Show that the product in multiplication and the dividend in division are corresponding terms. The multiplicand and the divisor. The multiplier and the quotient.

Exercises.

1. To 14 add 8; subtract 7; multiply by 5; multiply by 6; divide by 10.

2. From 32 take 17; add 25; multiply by 7; divide by 8; add 45; subtract 39; multiply by 7; divide by 14.

3. Multiply 24 by 6; divide by 12; add 69; subtract 17; multiply by 8; divide by 16; subtract 19; add 87.

4. Divide 91 by 7; multiply by 8; subtract 26; add 22; divide by 20; multiply by 23; subtract 38; add 31; divide by 12.

5. Divide 104 by 8; add 27; divide by 20; add 35; multiply by 9; subtract 33; divide by 6; add 32; subtract 28; multiply by 4; divide by 18; add 76; subtract 31; multiply by 9; divide by 27.

1. How many are 3 times 1? 5 times 1? 7 times 1? 2. How many ones in 3? In 5? In 7? In 11?

3. What two numbers multiplied together produce 5? 7? 4. What two numbers only can divide 5 without a remainder? What two only can exactly divide 7? 11?

5. What two numbers other than itself and 1 multiplied together produce 6? 10? 15? 21? 35?

6. What two numbers other than itself and 1 can divide 6 without a remainder? 10? 15? 21? 35?

7. Of what number are 2 and 3 the factors? 3 and 3? 2, 2, and 3? 2, 3, and 3? 3, 3, and 5? 3, 4, and 5? 8. Name all the numbers that can divide 12 without a remainder; 15; 18; 20; 24; 28; 30; 36.

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

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

Definitions.

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

Thus, 2 and 5 are factors of 10; 2, 3, and 5, of 30.

114. 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, etc.

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

Thus, 12 is a composite number, since 12 3 X 4, or 12 2 X 6.

116. A Prime Factor is a prime number used as a factor; and a Composite Factor is a composite number used as a factor.

Thus, in 3 X 4 = 12, 3 is a prime factor of 12, and 4 is a composite factor.

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

Thus, 3 is an exact divisor of 15; 5, of 20; 7, of 28; etc.

Note. The terms factor and divisor differ only in use: factor suggesting the process of multiplication, and divisor, the process of division.

11. Name all the prime numbers from 0 to 30.

12. Name all the composite numbers from 0 to 30. 13. Name the prime factors of 12; 18; 20; 24; 28. 14. What are the composite factors of 18? 24? 32? 36? 15. What are the smallest numbers other than 1 that can exactly divide 15? 24? 27? 35?

16. What are the largest numbers other than the number itself that can exactly divide 24? 28? 36? 42? 48?

17. What is the smallest and what is the largest prime factor of 42? Name all the composite factors.

18. Name all the different sets of two factors of which 24 is the product. Of which 36 is the product.

19. Name all the different sets of three factors of which 36 is the product. Of which 48 is the product.

20. What numbers from 40 to 60 are the product of two or more prime factors? Of two or more composite factors? Of a prime and a composite factor?

21. Show that a factor of any number is also a factor of any number of times that number.

22. Name all the different values of articles that can be bought for 48 cents. For 54 cents.

118. Principles.

1. 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.

IV. Any factor of a number is also a factor of any integral number of times that number.

Exercises.

Name all the different divisors of 24.

119. Model. The prime factors or divisors of 24 are 2, 2, 2, and 3. The composite divisors are 2 times 2, or 4; 2 times 3, or 6; 2 times 2 times 2, or 8'; and 2 times 2 times 3, or 12. Hence, all the divisors of 24 are 2, 2, 2, 3, 4, 6, 8, and 12.

1. Tell which of the following numbers are prime, and give the reason: 5, 7, 9, 14, 17, 18, 21, 23, 27, 29, 31, 35, 39, 41, 45, 49.

2. Tell which of the following numbers are composite, and give the reason: 6, 8, 11, 15, 17, 19, 25, 33, 37, 42, 51, 53, 57, 59, 61, 69.

Name the prime, and the composite factors, and all the different divisors of

1. 15, 18, and 20. |

4. 48, 54, and 56. | 7. $84, 88 feet, and 90 yards. 5. 64, 66, and 70. 8. $92, 96 tons, and 98 years. 3. 36, 40, and 45. 6. 75, 80, and 81. 9. $100,108 days, and 112 cords.

2. 24, 27, and 32.

2. Common Factors, or Divisors.

1. What are the factors of 15? Of 21? Of 33?

2. What are the exact divisors of 15? Of 21? Of 33? 3. What factors, or divisors, are common to 15 and 21? 4. Name a factor, or exact divisor, common to 18 and 24. 5. Name two numbers of which 7 is a common factor. 6. What prime factors have 18 and 27 in common?

7. What is the greatest factor that exactly divides 18 and 27? 24 and 36? 24, 32, and 40?

8. What factor, or exact divisor, is common to 7 and 3 times 7? To 9 and 5 times 9?

9. What factor, or divisor, is common to 6 and 9? To 6 and 9, and 54, their product?

10. What factor, or divisor, is common to 18 and 24, and to 42, their sum? To 18 and 24, and 6, their difference?

Definitions.

120. A Divisor, or Measure, is a number that divides any given number without a remainder.

Thus, 6 is a divisor of 24, since 6 divides 24 without a remainder. 121. A Common Divisor is a number that divides each of two or more given numbers without a remainder. Thus, 6 is a common divisor of 24 and 36, since it divides each of them without a remainder.

122. The Greatest Common Divisor is the greatest number that divides each of two or more given numbers without a remainder.

Thus, 12 is the greatest common divisor of 24 and 36, since it is the greatest number that divides each of them without a remainder.

123. Numbers are prime to each other when they have no common factors, or divisors.

Thus, 7 and 12 are prime to each other; also, 9 and 16; 9, 16, and 25.

11. What numbers are exact divisors of 12 feet? Of 16 feet? 12. Show that every common divisor of 12 and 16 is also a divisor of their sum, 28.

13. Show that a common divisor of any two numbers is also a divisor of their sum.

14. Show that every common divisor of 12 and 28 is also a divisor of their difference, 16.

15. Show that a common divisor of any two numbers is also a divisor of their difference.

16. Name four odd numbers of which 7 is the greatest common divisor. Four even numbers.

17. What divisors are common to 20 and 30? To their sum? To their difference? To their product?