PROB.-What is the greatest common divisor of 84 and 132?

FORM.*-1. By inspection, we find that the factors 2, 2, 3, are common to both 84 and 132. Since only common factors will divide both numbers, the product of all the common factors must be the greatest factor that will divide both numbers without a remainder.

2. Twelve is the product of all the common factors, 2, 2, 3, and is, therefore, the greatest common divisor of 84 and 132.

I. Rule. (a.) Resolve the numbers into their prime factors, take all the common factors, and their product will be the greatest common divisor.

II. Rule. (b.) Divide the numbers by any number that will divide all of them without a remainder; continue dividing each successive set of quotients until they have no common factor: the product of the divisors will be the greatest common divisor.

LESSON XII.

Find the greatest common divisor.

83. 85, and 95.

84. 72, and 168.

85. 119, and 121.

86. 12, 18, 24, and 30.
87. 14, 28, and 21.
93. I have three rooms

88. 20, 16, and 48.
89. 28, 16, 12, and 8.
90. 11, 88, and 99.
91. 28, 63, 47, and 93.

92. 78, 234, and 468. respectively 15, 18, and 24 feet in *NOTE. For other methods of finding greatest common divisor, see Commercia Arithmetic.

REVIEW.-What is division? (50.) What is the dividend? (51.) What is the divisor? (52.) What is the quotient? (53.) What is the difference between long division and short division? (55. 54.) What is a remainder in division? (56.) Illustrate the use of the sign of division. (57.) In what does the frac tional sign consist?

width; how wide can the oil-cloth be that will just fit in the three rooms.

LESSON XIII.

COMMON MULTIPLE.

119. A Multiple of a number is one that can be divided by it without a remainder.

120. A Common Multiple of two or more numbers is one that can be divided by each of them without a remainder.

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

What is the least common multiple of 6, 9, and 12 ?'

FORMULA.-1. The least number divisible by 12 must contain only the highest powers of all the different factors of 12, which are 23, and 3.

2. The least number divisible by 12 and 9 must contain only the highest powers of all the different factors of 12 and 9, which are 22 and 32. We reject the factor 3 in the number 12, because a higher power of 3 is found in the number 9.

3. Since the least number divisible by any number is also divisible by any factor of that number, therefore reject the factors of 6, for 6 is a factor of 12.

12.

Therefore 22×33-36 is the least common multiple of 6, 9, and

(b.) This abbreviated method of operation is generally preferred, and as the analysis is about the same as that of the other method, it will not be repeated.

* NOTE.-Reject the 6, for it is a factor of 12. Factors can be rejected before a formal division as well as by it.

(a.) Rule.-I. Resolve the numbers into their prime factors, and the product of the highest powers of all the different factors will be the least common multiple.

(6.) Rule.—II. Write the numbers in a horizontal line, leaving out all numbers that are factors of any of the other numbers; divide by the least prime number that will divide more than one of them without a remainder, and write the quotients and the undivided numbers below. Reject factors and continue to divide as before, until there is no prime number greater than one that will divide two or more of them without a remainder. The product of the divisors and the undivided numbers is the least common multiple required.

LESSON XIV.

EXERCISES FOR PRACTICE.

Find the least common multiples.

122. A Fraction is an expression denoting one or several of the equal parts of a number.*

kinds, Common and Decimal.

Fractions are of two

* NOTE-When the number is not mentioned unity is understood. Thus signifies of one.

123. A Common Fraction is one whose denomi nator is not ten nor a power of ten; as, 4, 3 †1, 185.

346

124. A Decimal Fraction* is one whose denominator is ten, or some power of ten; as, ‰, 185, 180, 18680. They are usually expressed by the use of the decimal point; as, .3, .04, .037, .00346.

(a.) A Common Fraction is expressed by a Numerator and a Denominator, which are called the terms of the frac tion.

125. The Denominator of a fraction shows the number of equal parts into which the unit is divided, and gives the name to the fraction. It may be written at the right of the numerator in words; as, 4 tenths, 9 sevenths, 8-11ths; or under the numerator with a short horizontal line between them;

126. The Numerator of a fraction shows how many of the equal parts into which a unit is divided are expressed. It may be written either at the left of the denominator; as, 3 tenths, 4 sevenths, or over the denominator with a short horizon. tal line between them; as, %, 4.

127. A Proper Fraction is less than a unit; as, 4, 1, 8.

128. An Improper Fraction is equal to or greater than a unit; as, 4, 1, 1, Y.

129. A Mixed Number is an integer with a fractional number added to it; as, 3+4, 7+. The sign of addition is usually omitted; as, 34, 7%.

130. A Simple Fraction is one whose numerator and denominator are both integers.

131. Complex Fractions are those which have frac

*NOTE-Decimal Fractions are often used in connection with common fractions, and are then written with a denominator in the same manner, and treated as com mon fractions; as, &× 70, +1809-180

tions or mixed numbers for their numerators or denominators,

132. A Compound Fraction is a fraction of a fraction. The parts are usually connected by the word of, or the sign of multiplication (×); as, of 121, § × 8, † of & × 3.

133. The Denomination of a fraction is determined by the number of equal parts into which the unit is divided; as, halves, fourths, &c.

134. The Value of a fraction of any denomination depends upon the number of equal parts expressed by the nume

rator.

135. The Reduction of fractions is the process of changing their denominations without changing their values.

136. Reduction Ascending is the process of changing a fraction to a higher denomination; as, fourths to halves, thirty-seconds to sixteenths, &c.

(a.) A Fraction is said to be expressed in its lowest terms (or highest denomination),* when its numerator and denominator are prime to each other.

137. Reduction Descending is changing a fraction to a lower denomination; as, halves to fourths, fifths to tenths, &c.

LESSON II.

MENTAL EXERCISES.

The teacher should by no means omit these mental exercises.

1. If a unit is divided into 3 equal parts, what are the parts called? If into 8 equal parts? 7? 11? 15? 18? 13? 26? 37? 95? 106? 38?

2. If a unit is divided into 75 equal parts, what will one of

* NOTE.-Let the teacher make it plain to the pupil that the lowest terms and the highest denomination of a fraction are the same.