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ORAL EXERCISE

1. State the prime factors of 15, 18, 27, 35, 42, 75. 2. State one factor of 395, 123, 777, 692, 1275, 1263. 3. Which of these are divisible by 3: 77, 609, 1203? 4. Which of these are divisible by 2: 68, 4973, 2870?

65. Other tests. There are various tests of divisibility besides those on page 32. They are easily illustrated or explained, and may be given or not as the teacher prefers. The more important are the following.

66. Divisibility by 4. A number is divisible by 4 if the number represented by the two right-hand figures is so divisible.

67. Divisibility by 6. A number is divisible by 6 if it is even and if the sum of its digits is divisible by 3.

68. Divisibility by 8. A number is divisible by 8 if the number represented by the three right-hand figures is so divisible.

69. Divisibility by 9. A number is divisible by 9 if the sum of its digits is so divisible.

70. Divisibility by 11. A number is divisible by 11 if the difference between the sums of the digits in its even and odd places is so divisible.

For example, 430,507 is divisible by 11, for 7 + 5 + 3 = 15, and 0 + 0 + 4 = 4, and 15 – 4 = 11.

71. Finding prime factors. Find the prime factors of 2310.

By § 62, 2 is a factor. By § 63, 5 is a factor of the other factor, 1155. By § 64, 3 is a factor of the other factor, 231. It is easy to see that 7 and 11 are factors of 77. Hence the prime factors are 2, 5, 3, 7, 11.

2)2310 5)1155

3)231 7)77

11

72. Finding the g.c.d. In the same way it is easy to find the greatest common divisor of numbers.

For example, to find the greatest common divisor of 231 and

660. Factoring 660 and 231 as here shown, we have

and

231 = 3 x 7 x 11,

660 = 2 × 2 × 3 × 5 × 11,

in which 3 and 11 are the only common factors. Therefore 3 × 11, or 33, is the greatest com

mon divisor.

3)231 2)660 7)77 2)330

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WRITTEN EXERCISE

Find the prime factors of the numbers in Exs. 1–30:

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Find the greatest common divisor in Exs. 31–48:

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ORAL EXERCISE

1. Name two multiples of 5; of 7; of 9; of 11.

2. Name a multiple that is common to 3 and 5.

3. The multiples 35 and 70 are common to 5 and 7. Which is the least multiple that is common to 5 and 7? 4. In the same way name the least common multiple of 2 and 3, and then name two other multiples.

73. Common multiple. A multiple of each of two or more numbers is called a common multiple of the numbers.

74. Least common multiple. Of all the common multiples of two or more numbers the least is called the least common multiple (1.c.m.).

For example, 24 is a common multiple of 4 and 6, but 12 is the least common multiple.

75. Finding the 1.c.m. The least common multiple of two numbers is easily found if their factors are known.

Thus, to find the least common multiple of 16 and 40. that

16 = 2 × 2 × 2 × 2, and 40 2 × 2 × 2 × 5.

We see

To be a multiple of 16, the least common multiple must contain 2 × 2 × 2 × 2, and to contain 40 it must also contain 5. Therefore 2 × 2 × 2 × 2 × 5, or 80, is the least common multiple.

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COMMON FRACTIONS

ORAL EXERCISE

1. When anything is divided into 3 equal parts, what is each part called? into 5 equal parts? into 12 equal parts?

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sixths? from B, that equals how many sixths?

3. How many sixths of the rectangle make the whole rectangle? How many thirds?

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How many halves?

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If A is called 1, what is B? C?

6. Each small rectangle is what part of the whole rectangle? Each black triangle is what part of the whole rectangle? What does this tell you about of ?

8

7. From the picture show that of the rectangle is of of it; also that3

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; also that

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}; also that

8. Name other fractions to which the following are equal:

8

15 18

f2, 12, 3, 71, 25, 4%, 14, 11, 18, 19, 13.

9 2

76. Unit. Any one thing is called a unit.

77. Fraction. One or more of the equal parts of a unit is called a fraction.

78. Denominator. The number which shows into how many equal parts a unit has been divided is, called the denominator.

For example, a sphere has been divided into 4 equal parts, and B is of it. Then

4 is the denominator.

A

B

C

We read ft. "one third of a foot."

79. Numerator. The number which shows how many parts have been taken to make a fraction is called the numerator.

In the fraction 2, 3 shows the number of fourths that have been taken, and is, therefore, the numerator.

80. Terms. The numerator and denominator together are called the terms of the fraction.

81. Common fraction. A fraction which has both terms expressed is called a common fraction.

For example, 3, §, 7, f, but not a decimal fraction like 0.6.

82. Proper fraction. A fraction whose numerator is less than the denominator is called a proper fraction.

For example, 3, 17, 180, 4.

83. Unit fraction. If the numerator is 1, the fraction is called a unit fraction.

For example, C is of a sphere.

84. Improper fraction. A fraction whose numerator equals or exceeds the denominator is called an improper fraction. For example, A is of a sphere, and A+ C is .

85. Mixed number. The sum of a whole number and a fraction is called a mixed number.

For example, if sphere A is 1, A + B is 13, A+ C is 14.

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