NOTE.--The work may sometimes be shortened by dividing by a composite number, remembering afterwards to substitute the factors of that number for the number itself. Thus, in the above we may divide by 9 instead of dividing by 3 twice.

100. A Common Divisor of two or more numbers is any number that will exactly divide each of them; thus, 2 is a common divisor of 12 and 18.

101. The Greatest Common Divisor is the greatest number that will exactly divide each of them; thus, 6 is the greatest common divisor of 12 and 18.

102. ILLUSTRATIVE EXAMPLE.

Find the greatest common divisor of 12, 30, and 42.

RULE I. To find the greatest common divisor of two of

more numbers: Separate the numbers into their prime factors and find the product of such as are common.

NOTE.-In Example 4, 18 is a factor of 36, and 12 of 48. The G.C.D. of 18 and 12 must be the G. C. D. of 18, 12, and their multiples, 36 and 48; .. we need only find the G. C. D. of 18, 12, and 42.

5. 42, 28, and 84.

6. 26, 52, and 65.

Find the G. C. D. of

7. 32, 18, 108, and 25. I
8. 114, 102, 78, and 66.

104. When numbers cannot readily be separated into their factors, the following method may be adopted :

ILLUSTRATIVE EXAMPLE.

OPERATION.

01) 325 (3 273

52) 91 (1

52

39) 52 (1

39

Find the G. C. D. of 91 and 325. We divide 325 by 91, to see if it is a divisor of 325, for 91 is the greatest divisor of itself; if it is a divisor of 325, it is the G. C. D. of 91 and 325. It is not a divisor of 325, for there is a remainder of 52. 52 is the greatest divisor of itself; if it is a divisor of 91, it is the G. C. D. of 52 and 91. It is not a divisor of 91, for there is a remainder of 39; 39 is the greatest divisor of itself; if it is a divisor of 52, it is the 00 G. C. D. of 39 and 52. It is not a divisor of 52, for there is a remainder of 13; 13 is the G. C. D. of itself and 39. It must therefore be of 39 and 52, for 52=1 × 39+ 13. If it is the G. C. D. of 39 and 52, it must be of 52 and 91, for 911 × 52+ 39. If it is the G. C. D. of 52 and 91, it must be of 91 and 325, for 325=3 × 91 +52. Hence the following:

13) 39 (3
39

RULE II. To find the G. C. D. of two numbers: Divide the greater number by the less, and the less number by the re mainder, if there is any, and thus proceed, dividing the last

*Greatest Common Divisor.

divisor by the last remainder, until nothing remains. The last divisor is the G. C. D. sought.

To find the G. C. D. of more than two numbers, find the of them, and then of that divisor and a on till all the numbers are taken.

G. C. D. of any two

third number, and so

15. What is the width of the widest carpeting that will exactly fit either of two halls, 45 feet and 33 feet wide respectively?

Ans. 3 ft.

16. A has a piece of ground 90 feet long and 42 feet wide, what is the length of the longest rails that will exactly suit its length and its width? Ans. 6 ft.

17. A lady has one flower bed measuring 10 feet around, &nd another measuring 18 feet. If she borders the beds with pinks, what is the greatest distance she can set her pink roots apart, and have them equally distant in the two beds?

Ans. 2 ft.

18. A man has 90 bushels Kidney potatoes, 60 bushels Jackson Whites, and 105 bushels Red Rileys. If he puts them all into the largest bins of equal size that will exactly measure either lot, how many bushels will each of his bins contain?

19. What is the length of the longest stepping-stones that will exactly fit 3 streets, 72, 51, and 87 feet wide, respectively?

20. What is the length of the longest curb-stones that will exactly fit 4 strips of sidewalk, the first being 273 feet long, the "econd 294, the third 567, and the fourth 651?

For Dictation Exercises, see Key.

r

FRACTIONS.

106. A Fraction is one or more of the equal parts of a unit; thus,, read three fourths, shows that a unit has been divided into four equal parts, and that three of those parts are taken.

107. The number which shows into how many equal parts unit is divided, is called the Denominator of the fraction, because it denominates or names the parts; thus, 4 is the denominator of 2.

108. The number which shows how many parts are taken, is called the Numerator; thus, 3 is the numerator of 2.

109. The numerator and denominator are called the Terms of a fraction.

110. A Common or Vulgar Fraction is a fraction whose denominator and numerator are both expressed, the numerator being written above, and the denominator below, a dividing line; as,,, 25.

111. A Decimal Fraction is one whose denominator is 10, or some integral power of 10. The denominator is not generally expressed. written .2, and written .36, are decimal fractions.

36

112. A Mixed Number is a whole number and a fraction expressed together, as 78, 213.

113. Common fractions may be either Proper, Improper, Compound, or Complex.

114. A Proper Fraction is one whose numerator is less than its denominator, as 4.

115. An Improper Fraction is one whose numerator equals or exceeds its denominator, as 33, 1.

116. A Compound Fraction is a fraction of a fraction, as { of §.

117. A Complex Fraction is one which contains a fraction In either or both of its terms, as 2§, 8

QUESTIONS. What does the denominator of a fraction show? What does the numerator show?

What is meant by the expression ? Ans. 5 of the 8 equal parts into which a unit is divided.

What is meant by the expression ?? 18? 18?

118. If we compare common fractions with the last expression for division in Art. 55, we shall see that their forms are alike. A fraction implies division, the numerator being the dividend, and the denominator the divisor. Thus, may be considered either three fourths of 1 or one fourth of 3. The following diagram will show that these are equivalent expressions, † of the one line in figure 1 being equal to of the three lines in figure 2.

What does

of 3 = 2.

denote? Ans. It denotes either 5 of the seven equal parts into which 1 is divided, or one seventh of 5. What does

show ? ? 18?

119. GENERAL PRINCIPLES.

NOTE.-The following propositions should be copiously illustrated by the teacher, and frequently referred to, until they are fully comprehended by the pupil.

PROPOSITION I. As the denominator of a fraction shows the number of parts into which a unit is divided, and the numerato shows how many parts are taken, it follows that if we multiply the numerator of a fraction by a whole number, we multiply thes number of parts, and thus increase the value of the fraction; but if we multiply the denomind or of a fraction, we multiply the number of parts into which a unit is divided, and thus diminish the size of the parts, and consequently decrease the value of the fraction.