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CHAPTER III

FACTORS AND MULTIPLES

DEFINITIONS

78. An integer (Latin integer, whole) is a whole number; as, 1, 2, 4, etc.

NOTE.-Only integral numbers are considered here. Integral numbers are either prime or composite.

79. A prime number (Latin primus, first) is a number that cannot be divided by any integer (except itself and 1) without a remainder; as 1, 2, 3, 5, 7, etc.

80. A composite number (Latin componere, to put together) is a number that can be exactly divided by some other integer than itself and 1; as, 4, 6, 9, etc.

Integral numbers are either even or odd.

81. An even number (Old English efen, level) is a number exactly divisible by 2. Even numbers always end in 0, 2, 4, 6, or 8.

82. An odd number (Old English odde, without a mate) is a number which, when divided by 2, has 1 for a remainder. Odd numbers always end in 1, 3, 5, 7, or 9.

83. A factor of a number (Latin factor, a maker) is any exact divisor of the number.

The factors of a number, when multiplied together, make the number.

NOTE.-In giving the factors of a number, it is customary to omit 1 and the number itself.

84. A prime factor is a prime number that is a factor. 85. A common factor of two or more numbers is a number that will exactly divide each of them.

86. The highest common factor (h. c. f.) of two or more numbers is the highest number that will exactly divide each of them.

NOTE. The terms, divisor, factor, measure, and submultiple, are commonly used to mean the same thing.

87. A multiple of a number (Latin multus, many, and plus, more) is any integral number of times the number. Thus, 16 is a multiple of 4, because it is a number of times 4.

88. A common multiple of two or more numbers is a number that will exactly contain each of them.

89. The lowest common multiple (1. c. m.) of two or more numbers is the lowest number that will exactly contain each of them.

TESTS OF DIVISIBILITY

90. Illustrate each of the following statements:

1. 2 is a factor of any even number.

2. 3 is a factor of any number the sum of whose digits
is divisible by 3.

3. 4 is a factor of a number if the number expressed
by its two right hand figures is divisible by 4.
4. 5 is a factor of any number whose right hand figure
is 0 or 5.

5. 6 is a factor of any even number the sum of whose
digits is divisible by 3.

6. 8 is a factor of a number if the number expressed by its three right hand figures is divisible by 8. 7. 9 is a factor of any number the sum of whose digits is divisible by 9.

8. 10 is a factor of any number whose right hand figure is 0.

9. 11 is a factor of a number if the sum of the digits in the odd places minus the sum of the digits in the even places is 0 or a multiple of 11. 24629 is divisible by 11, for (9+6+2) − (2+4)=11.

I. HIGHEST COMMON FACTOR

91. The highest common factor of two or more numbers is commonly called the greatest common divisor (g. c. d.). 92. Fundamental principles used in finding the h. c. f.: 1. A factor of a number is a factor of any multiple of that number.

2. A factor of each of two numbers is a factor of their

sum.

3. A factor of each of two numbers is a factor of their

difference.

93. The method of factoring may be illustrated by the following

Example: Find the h. c. f. of 24, 36, 48, and 60.

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1. Since 2 is found twice in each, the h. c. f. must contain two 2's.

2. Since 3 is found once in each, the h. c. f. must con

II. tain one 3.

3. Since no other factors are common, the h. c. f. must contain no other factors.

4. Therefore, the h. c. f. of 24, 36, 48, and 60 = 12.

With small numbers, this method is preferable. With large numbers, the long division method is preferable. This method is sometimes called the Euclidean method, from Euclid, who used it about 300 B.C.

94. The Euclidean or long division method may be illustrated by the following

Example: Find the h. c. f. of 216 and 324.

OPERATION.
216)324(1

216
108)216(2

216

EXPLANATION.-If 216 were the h. c. f., it would be necessary for it to be contained in 324 at least twice, and in their difference at least once. Since this is not the case, 216 is not the h. c. f. ciple 3, 92, the h. than 108; therefore, we take 108 and 216. must also be a factor of 216, which it is. h. c. f. of 216 and 324.

According to princannot be greater If 108 is the h. c. f., it

c. f.

108 is therefore the

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Using long division, find the h. c. f. of:

7. 160, 256, and 1600. 8. 244, 1220, and 732.

9. 59, 295, and 177.

11. 175, 250, and 625.

10. 765, 855, and 1035.

12. 187, 921, and 888.

II. LEAST COMMON MULTIPLE

95. Fundamental principles to be observed:

1. The 1. c. m. of two or more numbers cannot be less than the largest of the numbers; that is, it must contain all the prime factors of the largest of the numbers.

2. It must contain all the prime factors of the smaller numbers which are not factors of the largest.

3. It must contain no other factors.

96. The method of factoring may be illustrated by the following

Example: Find the 1. c. m. of 36, 72, and 108.

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According to the first principle above, we must use all the factors of 108. Since the factor 2 is found in 72 one time more than in 108, we must use this factor again. But according to principle 3, no other factors must be used. Therefore, the 1. c. m. is 2×2×2×3×3×3=216.

97. The highest common factor may be used in finding the least common multiple. Using the example above: 1. The h. c. f. of 36, 72, and 108 is 36.

2. This factor is used once and no more in the 1. c. m.

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