18. What is the unit of 14 feet? common divisor of 14 feet and 21 feet?

Of 21 feet? Of the

19. Name four numbers of which 3 and 9 are common divisors. Four of which 9 is the greatest common divisor.

20. Tell every number that is a common factor of 18 yards and 24 yards. Of 27 feet and 36 feet.

21. What is the length of the longest pole that will exactly measure 20 feet, 30 feet, and 40 feet?

22. What is the greatest number of men among whom $24, $32, and $40 could be exactly divided, so that each should have a whole number of dollars?

124. Principles.

I. A factor common to two or more numbers is a common divisor of those numbers.

II. The only common divisors of two or more numbers are their common prime factors, and the product of any two or more of them.

III. The greatest common divisor of two or more numbers is the product of all their common prime factors.

IV. The common divisor of two numbers is also a divisor of their sum and of their difference.

Exercises.

Name the common prime divisors, the common composite divisors, and the greatest common divisor of 24 and 36.

125. Model. The prime divisors of 24 are 2, 2, 2, and 3; the prime divisors of 36 are 2, 2, 3, and 3. The common prime divisors of 24 and 36 are 2, 2, and 3. The common composite divisors of 24 and 36 are 2 times 2, or 4; 2 times 3, or 6; and 2 times 2 times 3, or 12. The greatest common divisor of 24 and 36 is the product of all their common prime divisors, 2, 2, and 3, or 12.

Name the common prime divisors, the common composite divisors, and the greatest common divisor of

3. Common Multiples, or Dividends.

1. What number is the product of the factors 3 and 5? 2. What number can be exactly divided by 3 and 5? 3. Name three products of which 3 is a factor; 5; 7. 4. What are the prime factors of 6 and 2 times 6? Of 8 and 3 times 8? Of 9 and 5 times 9?

5. Of what two numbers is 21 the product? By what two numbers can 21 be exactly divided?

6. Name some product of which 6 and 9 are common factors. Some dividend of which 6 and 9 are common divisors.

7. What is the least number of which 6 and 9 are common factors? The least number of which they can be exact divisors?

8. Name the prime factors of 10 and of 15. What is the product of the different prime factors of 10 and 15?

9. What are the prime factors of the least number that can be exactly divided by 10 and 15?

10. What is the least number of which 4, 6, and 8 are common factors? The least number of which they are common divisors?

Definitions.

126. A Multiple, or Dividend, is a number that can be divided by any given number without a remainder. Thus, 15 is a multiple, or dividend, of 5, since 5 is a factor, or divisor, of 15.

127. A Common Multiple is a number that can be divided by each of two or more numbers without a remainder.

Thus, 15 is a common multiple of 5 and 3; so, also, is 30.

128. The Least Common Multiple is the least number that can be divided by each of two or more numbers without a remainder.

Thus, 12 is the least common multiple of 3 and 4, or of 2 and 6, since it is the least number that can be exactly divided by 3 and 4, or by 2 and 6. Note. The terms multiple and dividend differ only in use: multiple suggesting the process of multiplication, and dividend, of division.

11. Name three multiples of 9 feet. Of 12 feet. 12. Show that a number can have any number of multiples. 13. Name three common multiples of 9 feet and 12 feet. 14. Show that two or more numbers may have any number of common multiples.

15. Name the least common multiple of 9 feet and 12 feet. 16. Show that two or more numbers can have but one least common multiple.

17. What is the unit of 10 yards? Of 15 yards? Of their least common multiple?

18. Name two numbers of which 6 and 8 are factors. The least number of which they are common factors.

19. What number is a common multiple of 5 quarts, 6 quarts, and 8 quarts?

20. What is the smallest sum of money that can be made up of 2-cent, 3-cent, 5-cent, or 10-cent pieces?

21. What is the narrowest box in which ribbons 2 inches wide, 3 inches, or 4 inches could be packed?

22. What is the smallest sum of money for which you could buy a number of oranges at 5 cents, 8 cents, or 10 cents each?

129. Principles.

I. A multiple of a number contains all the prime factors of that number.

II. A common multiple of two or more numbers contains all the prime factors of each of those numbers.

III. The least common multiple is the least number that contains all the prime factors of each of two or more numbers.

Exercises.

Name two common multiples, and the least common multiple of 9 and 12.

130. Model. - Since the common factors of a number are also common divisors of that number, a common multiple of 9 and 12 is any number of which they are factors, and they are factors of their product, 9 times 12, or 108; also, of twice 108, or 216; etc.

The prime factors of 9 are 3 and 3; the prime factors of 12 are 2, 2, and 3. The least common multiple of 9 and 12 is the least number that is the product of the different prime factors of 9 and 12, or the product of 2, 2, 3, and 3, which is 36.

Name two common multiples, and the least common multiple of

1. If a cake is cut into two equal parts, what is each part called? What part of the cake is each part?

2. Into how many halves can an apple be cut? How many halves are in a unit, or in anything?

3. If an apple is divided into four equal parts, what name is given to one of the parts? How many fourths in one?

4. What is one of the six equal parts of a unit named? What are two of the parts named? How many sixths in a unit?

5. Into how many halves can a unit, or anything, be divided? Into how many thirds? Fifths?

Sevenths?

6. How many fourths are there in a unit? How many sixths? How many eighths? Tenths?

7. What is meant by one half of a unit? By one fourth? One sixth? One fifth? One seventh? One ninth?

8. What is meant by two thirds of a unit? By three fourths? Three fifths? Five sixths?

9. Which is the greater, one half or one fourth of a unit? One third, or one sixth? One fourth, or one eighth?

10. Which are the smaller parts of a unit, halves or thirds? Fourths, or fifths? Sixths, or eighths? Ninths, or tenths?

Definitions.

131. A Fraction is one or more of the equal parts of a unit. .

Thus, 1 fourth, 2 fifths, 3 eighths are fractions.

132. Fractions are divided into two classes: Common Fractions and Decimal Fractions.

133. A Common Fraction is expressed in figures by two numbers written one above the other, with a line between them.

Thus,

134. The two numbers used to express a fraction are called the Denominator and the Numerator.

135. The Denominator or namer of a fraction is the number written below the line.

It shows into how many equal parts the unit is divided, and denominates or gives name to the parts.

Thus, 8 is the denominator of the fraction; it shows that the unit is divided into 8 equal parts, which are named eighths.