103. A General Principle on Common Factors. If a number is a common factor of two numbers, it is a factor of their sum and also of their difference.

Thus, 8 is a common factor of 24 and 56 and therefore of 56+ 24 = 80 and of 56 24 32.

This principle may be used in finding common factors.

Example. Find the G. C. D. of 1292 and 1615.

Solution.

1615

4

1292

1292 323)1292

323

If these numbers have a common factor, it is a factor of their difference, that is, of 323. By dividing we find 323 to be a factor of 1292 and of 1615. Hence 323 is the required G. C. D.

The general application of this principle is too complicated for this book and is very seldom if ever required in business practice.

104. Multiples. The product obtained by multiplying a given. number by an integer is called a multiple of the given number.

Thus, 6, 12, 18, 24 are multiples of 6. These numbers are also multiples of 2 and 3. Every integer is a multiple of 1.

105. Rule on Multiples. — If one of two numbers is a multiple of the other number, then every multiple of the first number is a multiple of the second.

That is, every multiple of 16 is a multiple of 2, 4, and 8; and every multiple of 60 is a multiple of 2, 3, 5, 10, 20, and 30.

106. Common Multiples. A number which is a multiple of each of two or more numbers is called a common multiple of these numbers. That is, 30 is a common multiple of 5, 6, 10, 2, 3.

ORAL EXERCISES

Find two common multiples of each of the following sets of

107. Least Common Multiples. The smallest number which is a common multiple of two or more numbers is called the Least Common Multiple of these numbers. The Least Common Multiple is usually denoted by L. C. M.

Thus the L. C. M. of 10, 15, 20 is 60, and the L. C. M. of 8, 16, 24 is 48. Finding the L. C. M. of numbers is of importance in dealing with fractions.

Example. Find the L. C. M. of 18, 32,

48.

We factor each number into its prime factors, as shown.

The L. C. M. of 18, 32, 48 must be a multiple of 48. That is, it must contain the factors 2, 2, 2, 2, 3.

Since the L. C. M. is a multiple of 32 it must contain an additional factor 2, and since it is a multiple of 18, it must contain an additional factor 3. Hence the L. C. M. of 18, 32, 48 is 2 × 2 × 2 × 2 × 2 × 3 × 3 = 288.

This problem may also be solved as follows:

2)18, 32, 48

3)9, 16, 24

8)3, 16, 8

3, 2, 1

First divide each number by 2. Then divide 9 and 24 by 3, bringing down the 16. Finally divide 16 and 8 by 8, bringing down the 3.

The L. C. M. is 2 × 3 × 8 × 3 × 2 = 288.

The process of division ends when no two numbers below the last line contain a common factor. The L. C. M. is the product of the numbers below the last line and of the factors to the left.

At each step divide by the largest number possible.

In finding the L. C. M. of the numbers 18, 24, 36, 48, we need only to find the L. C. M. of 36 and 48, since 18 and 24 are contained in these. For the purpose of finding the L. C. M. of a set of numbers any number which is contained in one of the others may be rejected.

EXERCISES

Find the L. C. M. of each of the following sets of numbers. In each case, state what numbers may be rejected.

MISCELLANEOUS WORK

1. Loads of grain were hauled to an elevator. Find the net weight in pounds if the gross weights and tares were as follows:

In each of the following find by how much the first number exceeds the sum of the numbers below it.

In each of the following which column has the greater sum and how

10. Give short cuts for multiplying by 33, 25, 50, 125, 369.

11. Give short cuts for multiplying 79 by 894, 96 by 98, 103 by 102.

12. Give short cuts for dividing by 63, 25, 50, 125.

13. Describe the testing of multiplication and division by the process of casting out 9's.

CHAPTER VIII

FRACTIONS, CANCELLATION

108. Use of Fractions in Business. - Simple fractions occur frequently in business, and no one is competent to do the figuring required in everyday transactions unless he understands fractions.

109. Fractions. A number in the form is called a fraction. There are several definitions of fractions in general use.

Thus may be regarded as one of the 3 equal parts of 2, or as 2 of the 3 equal parts of 1.

In more advanced works on arithmetic, it is usual to define as a number such that 3 X = 2.

110. Numerator, Denominator. A fraction is always written by means of two numbers (usually integers), one above a horizontal line, and the other below it. The number above the line is called the numerator, and the one below it, the denominator.

Every fraction is, therefore of the form:

numerator denominator

The denominator states into how many equal parts a unit has been divided; and the numerator states how many of these parts are represented by the fraction.

Thus, in the fraction the denominator 5 states that a unit has been divided into five equal parts and the numerator 4 states that four of these parts are represented by the fraction.

The numerator and denominator of a fraction are called the terms of the fraction.

A fraction may also be regarded as an indicated quotient, the numerator being the dividend, and the denominator, the divisor. Thus the fraction may be regarded as the quotient when 4 is divided by 5. A problem in division is often represented in the form of a fraction. To deal effectively with fractions it is now necessary to study them more in detail.

111. A General Property of Fractions. From the figure it is

1/8 1/8 1/8 1/8 1/8 1/8

1/8 1/8

1/4

1/4

1/4

1/4

1/2

1/2

1

These are instances of a general property of fractions:

Multiplying both terms of a frac

tion by the same number does not change its value.

Reading the above equations backward, we have

송

} = }, } = }, } = 1 = 1.

The above property may then be stated:

Dividing both terms of a fraction by the same number does not change its value.

These two rules may be stated in one as follows:

Multiplying or dividing both terms of a fraction by the same number does not change its value.

112. Fractions in Lowest Terms.

When the numerator and de

nominator of a fraction have no common factor except 1, the fraction is said to be in its lowest terms.

113. Proper Fractions. A fraction whose numerator is less than its denominator is called a proper fraction.

114. Improper Fractions. A fraction whose numerator is not less than its denominator is called an improper fraction.

Thus,,,are proper fractions, while,, are improper fractions.

terms may be reduced to its lowest terms by dividing both numerator and denominator by their greatest common factor.