105. Written Exercises.

58. What is the least common multiple of 20, 24, and 36?

than 360 will contain 20, 24, and 36, for if one of the 2's in the common multiple were omitted, it would not contain 24; if one of the 3's, it would not contain 36; and if the 5 were omitted, it would not contain 20.

Similar reasoning applies to all examples. Hence, to find the least common multiple of two or more numbers,

Rule 1.

Separate each number into its prime factors, and the continued products of all the different prime factors, each taken the greatest number of times it occurs in any of the given numbers, will be the least common multiple.

59. What is the least common multiple of 12, 16, 20, and 30? Ans. 2 X2 X2 X2×3×5=240.

60. What is the least common multiple of 22, 33, and 55? 61. What is the least common multiple of 16, 36, 40, and 48?

NOTE 1. As 48 contains 16, the least common multiple is not changed by omitting the 16; and the question is reduced to,- What is the least common multiple of 36, 40, and 48?

62. What is the least common multiple of 20, 30, 50, and 80? 63. What is the least common multiple of 15, 25, 45, and 75?

NOTE 2. As 75 contains both 25 and 15, the least common multiple is not changed by omitting 15 and 25. Apply this principle whenever you can.

64. What is the least common multiple of 35, 50, 75, and 901

65. What is the least common multiple of 24, 36, 48, and 64 ? 66. What is the least common multiple of 72, 80, 84, and 96? 67. What is the least common multiple of 42, 49, 72, and 88? 68. What is the least common multiple of 12, 20, 24, and 36?

106. The same result is sometimes more easily obtained by

Rule 2.

Having written the given numbers in a line, divide by any prime number that will divide two or more of them, and write the quotients and undivided numbers in a line beneath; proceed with this line as with the first, and so continue until no two of the numbers can be divided by any number greater than one; the continued product of the divisors and numbers in the last line will be the multiple sought.

The second rule may be illustrated by the example already employed in explaining the first rule, viz. :

69. What is the least common multiple of 20, 24, and 36?

OPERATION.

2) 20, 24, 36

2) 10, 12, 18

3) 5, 6, 9
5, 2, 3

Ans. 2×2×3×5×2×3=360.

If the process by the 1st rule be examined it will be seen that the factor 2 is found 7 times in the given numbers, and as 2 is taken but 3 times in finding the multiple, it is

rejected 4 times. By the 2d rule, also, 2 is rejected 4 times, viz. twice in the 1st division by 2, and twice in the 2d division by 2. The learner may think 2 is rejected three times in each of the first two divisions; but he must remember that the divisor 2 is retained as a factor in the common multiple in each instance.

NOTE 1. This second method is only another way of rejecting the common factors.

70. What is the least common multiple of 5, 16, 24, 32,

NOTE 2. As 16 is contained in 32, and 24 in 48, we omit 16 and 24 in the operation.

71. What is the least common multiple of 15, 35, 45, and 210 ? 72. What is the smallest sum of money with which I can buy horses at $75 each, cows at $50 each, or sheep at $9 each, using the same sum in each case? Ans. $450.

73. I have 4 wine measures; the first holds 4 liters, the second 5 liters, the third 6 liters, and the fourth 8 liters; what is the size of the smallest cask that can be exactly measured by means of each of these measures? Ans. 120 liters.

74. What is the least common multiple of 10, 15, 45, 75, and 90?

NOTE 3. If the numbers are mutually prime, their product is their least common multiple.

75. What is the least common multiple of 7 and 11?

Ans. 7 x 11=77.

76. What is the least common multiple of 4, 9, and 29?

NOTE 4. The least common multiple of two numbers is equal to their product divided by their greatest common divisor. This is only another way of rejecting common factors.

77. What is the least common multiple of 12 and 20?

The greatest common divisor of 12 and 20 is 4, and
The least common multiple is 12 × 20÷4=60, Ans.

78. What is the least common multiple of 36 and 45? 79. What is the least common multiple of 44 and 66 ?

CANCELLATION.

107. Cancellation is the striking out of equal factors from the dividend and divisor.

In this operation the factor 5 was cancelled from 1155 and 105, leaving 231 as the dividend and 21 as the divisor; then 3 was cancelled from 231 and 21, leaving 77 as the dividend and 7 as the divisor; then 7 was cancelled from 77 and 7, leaving 11 as the dividend and 1 as the divisor. Thus, all the common factors are cancelled from the dividend and the divisor, leaving 11 to be divided by 1. Hence,

To perform examples by cancellation,

Rule.

Cancel all the factors common to both dividend and divisor, then complete the required division.

NOTE 1. In operations involving both multiplication and division very much time and labor are saved by cancellation.

81. Divide 44 × 8 × 7 by 11 × 16 × 7.

Ans. 2.

82. Divide 40 x 18 x 13 x 8 by 10 x 13 x 16.

83. Divide 376 × 14 × 21 by 7 × 7 × 16 × 3.

84. Divide 5 × 25 × 874 by 2 × 437 × 5 × 5 × 5.

Ans. 1.

NOTE 2. When a number is entirely cancelled, 1 remains, though it need not

be written except when the dividend or divisor is cancelled completely.

85. Divide 108 × 17 × 9 × 4 by 27 × 3 × 16 × 17.

86. Divide 87846 by 7986.

87. Divide 15 × 4 × 8 × 9 by 30 × 2 × 6 × 12.

85. When is one number divisible by another? What is an exact divisor? When is one number indivisible by another? 86. What is an Even Number? 87. What is an Odd Number? 88. A Prime Number What is the only even prime number? 89. When are numbers mutually prime? 90. What is a Composite Number? 92. What are the Factors of a number? Is a number a factor of itself? 93. What are the prime factors of a number? 94. What is it to factor a number? What numbers are divisible by 2? By 3? 4? 5? 6? What is said of 7? What numbers are divisible by 8? By 9? By 10? 95. Give the Rule for finding the Prime Factors of a number. 97. What is a Common Divisor? 98. Greatest Common Divisor? 99. Rule 1 for finding the Greatest Common Divisor? In finding the Greatest Common Divisor of several numbers, what numbers may be neglected? 100. Rule 2? When do we use Rule 2? How is Ex. 47 solved? 102. What is a Multiple of a number? 103. A Common Multiple of two or more numbers? 104. The Least Common Multiple? 105. Rule for finding the Least Common Multiple? Reason? In finding the least common multiple of several numbers, how may the work often be abbreviated? 106. Rule 2 for finding the Least Common Multiple? 107. What is Cancellation? On what principle does it depend? Rule for Cancellation? What is the advantage of cancellation? When all the numbers are cancelled, what remains?