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Tell which of the following numbers are prime: 5. 11; 12; 13; 14; 15; 16; 17; 18; 19; 20. 6. 41; 43; 47; 49; 55; 59; 63; 31; 39; 81.

7. Name all the composite numbers from 30 to 50 inclusive.

8. Name the prime factors of 12; 18; 21; 24; 25; 27; 28; 30; 33.

9. Name the odd numbers to 20.

Divisors

53. A number that will divide a given number without a remainder is called an exact divisor or a divisor of the given number.

54. A divisor of each of two or more numbers is called a common divisor of them.

55. The greatest number that will divide each of two or more numbers is called their greatest common divisor (g. c. d.). 56. Some easy tests of divisibility:

A number is divisible:

by 2, if it ends in 0, 2, 4, 6, or 8.

by 3, if the sum of its digits is divisible by 3.
by 4, if the number expressed by the two right-
hand digits is divisible by 4.

by 5, if it ends in 0 or 5.

by 6, if it is divisible by 2 and by 3.

by 9, if the sum of its digits is divisible by 9.

by 12, if it is divisible by 3 and by 4.

by 15, if it is divisible by 3 and by 5.
by 25, if it ends in 00, 25, 50, or 75.

57. If a number is divided by one of its prime divisors, the quotient by one of its prime divisors, the next quotient by one of its prime divisors, and so on until the quotient becomes 1, these successive divisors are called the continued prime divisors of the number.

2|210

3 105

5 35

7 7

1

Thus, 2, 3, 5, and 7 are the continued prime divisors of 210.

58. The continued prime divisors of a number are the prime factors of that number.

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59. The continued common prime divisors of two or more numbers are their common prime factors.

2 84 126 210
342 63 105

714

21 35 2 3 5

Thus, 2, 3, and 7, the continued common prime divisors of 84, 126, and 210, are the common prime factors of 84, 126, and 210.

60. Any of the common prime factors of two or more numbers, or the product of any two or more of such factors, is a common divisor of the numbers.

Thus, the common prime factors of 84, 126, and 210 are 2, 3, and 7; hence, 84, 126, and 210 are each divisible by 2, 3, 7, 2 × 3, 2 × 7, 3 × 7, and 2 × 3 × 7.

61. The product of all the common prime factors of two or more numbers is their greatest common divisor.

Thus, the greatest common divisor of 84, 126, and 210 is 2 × 3 × 7.

Exercise 15

1. Name at sight two exact divisors of 54; 63; 90; 72; 56; 39; 66.

2. Name a common divisor of 12 and 18; 54 and 63; 72 and 96; 110 and 120.

3. Name a common divisor of 12, 18, and 24; 42, 49, and 56; 54, 63, and 72.

Tell at sight which of the numbers 2, 3, 4, 5, 6, 9, 12, 15, and 25, will divide:

4. 84 IO. 230

5. 99

II. 324

6. 105 7. 75 8.78 9. 146 12. 425 13. 423 14. 552 15. 627

Illustrative Example

Find the greatest common divisor of 90, 120, and 240.

290 120 240 345 60 120

5 15 20

40

3 4

8

Explanation:

2, 3, and 5 are the common prime factors of 90, 120, and 240.

The product of all the common prime factors of two or more numbers is their greatest common divisor.

.. 2 × 3 × 5, or 30, is the g. c. d. of 90, 120, and 240.

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62. A number that can be exactly divided by a given number is called a multiple of the given number.

63. A number that can be exactly divided by each of two or more numbers is called a common multiple of them.

64. The least number that can be exactly divided by each of two or more numbers is called their least common multiple (1.c. m.).

65. The least common multiple of two or more numbers is the product of all their different prime factors, each taken the greatest number of times that it occurs in any one of them.

Illustrative Examples

1. Find the least common multiple of 18, 24, and 36.

18=2 x 3 x 3

24 2 x 3 x 2x2

36=2 x 3 x3x2

Explanation:

The different prime factors of 18, 24, and 36 are 2 and 3. The greatest number of times that 2 occurs as a factor in

any of the numbers is three; the greatest number of times that 3 occurs as a factor in any of the numbers is two. ..2x2x2 x3 x 3, or 72, is the 1. c. m. of 18, 24, and 36.

2. Find the 1. c. m. of 120, 168, and 180.

2 120 168 180

23252

60 84 90

30

20 28

10

14

15

2

14 3

1

7

3

Explanation:

2, 3, and 2 are common prime factors of the numbers. 5 is a common prime factor of 120 and 180; 2 again a common prime factor of 120 and 168; 3 again a prime factor of 180; and 7 a prime factor of 168.

The different prime factors are 2, 3, 5, and 7. The greatest number of times that 2 occurs as a factor is three in 120; that 3 occurs is two in 180; that 5 occurs is once in either 120 or 180; that 7 occurs is once in 168.

.. 2×2×2×3×3×5×7, or 2520,= 1. c. m. of 120, 168, and 180. NOTE. When the numbers are easily factored, the first form is preferable

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