MULTIPLES. 105. A Multiple is a number exactly divisible by a given number. Thus, 20 is a multiple of 4; 50 is a multiple of 10; 360 is a multiple of 90; 10000 is a multiple of 10. 106. A Common Multiple is a number exactly divisible by two or more given numbers. Thus, 20 is a common multiple of 2, 4, 5, and 10; 60 is a common multiple of 2, 3, 5, 6, 10, and 15. The Least Common Multiple is the least number exactly divisible by two or more given numbers. Thus, 24 is the least common multiple of 3, 4, 6, and 8; 30 is the least common multiple of 2, 3, 5, 6, 10, and 15. 107. From the definition (106) it is evident that the product of two or more numbers, or any number of times their product, must be a common multiple of the numbers. Hence, A common multiple of two or more numbers may be found by taking the product of the given numbers. 108. By a little practice the multiples and least common multiples of the smaller numbers may be readily found by inspection. Since all even numbers are multiples of 2, the multiples of 2 are the most numerous. The following table presents the multiples from 1 to 100 of the prime numbers from 1 to 11: FACTORS. 235 7 11 MULTIPLES. All even numbers to 100. 15, 30, 45, 60, etc., to 90. 33, 66, 99. 5, 7 35, 70. The least common multiple of numbers mutually prime is always the least common multiple of their product. EXAMPLES. 109. To find the least common multiple. FIRST METHOD. From the nature of prime numbers we derive the following principles: I. If a number exactly contains another, it will contain all the prime factors of that number. II. If a number exactly contains two or more numbers, it will also contain all the prime factors of those numbers. III. The least number that will exactly contain all the prime factors of two or more numbers, is the least common multiple of those numbers. 1. Find the least common multiple of 30, 42, 66, and 78. 2 x 3 x 13 x 11 x 7 x 5 = 30030, Ans. SOLUTION.-The number cannot be less than 78, since it must contain 78; hence it must contain the factors of 78, viz.: 2 x 3 x 13. We here have all the prime factors of 78, and also all the factors of 66 except the factor 11. Annexing 11 to the series of factors, 2 × 3 × 13 x 11, we have all the prime factors of 78 and 66, and also all the factors of 42 except the factor 7. Annexing 7 to the series of factors, 2 × 3 × 13 x 11 x 7, we have all the prime factors of 78, 66, and 42, and also all the factors of 30 except the factor 5. Annexing 5 to the series of factors, 2 × 3 × 13 x 11 x 7 x 5, we have all the prime factors of each of the given numbers; and hence the product of the series of factors is a common multiple of the given numbers, (II). And as no factor of this series can be omitted without omitting a factor of one of the given numbers, the product of the series is the least common multiple of the given numbers, (III). RULE.-I. Resolve the given numbers into their prime factors. II. Take all the prime factors of the largest number, and such prime factors of the other numbers as are not found in the largest number, and their product will be the least common multiple. When a prime factor is repeated in any of the given numbers, it must be used as many times, as a factor of the multiple, as the greatest number of times it appears in any of the given numbers. Find the least common multiple of: 110. 1. What is the least common multiple of 4, 6, 9, and 12? OPERATION. 214 6 9 12 22 3 9 6 3 3 9 3 3 3 2 × 2 × 3 × 3 = 36, Ans. SOLUTION. We first write the given numbers in a series, with a vertical line at the left. Since 2 is a factor of some of the given numbers, it must be a factor of the least common multiple sought. Dividing as many of the numbers as are divisible by 2, we write the quotients and undivided number, 9, in a line underneath. We now perceive that some of the numbers in the second line contain the factor 2; hence the least common multiple must contain another 2, and we again divide by 2, omitting to write down any quotient when it is 1. We next divide by 3 for a like reason, and again by 3. By this process we have transferred all the factors of each of the numbers to the left of the vertical line; and their product, 36, must be the least common multiple sought, (109, III). 2. What is the least common multiple of 10, 12, 15, SOLUTION. We readily see that 2 and 5 are among the factors of the given numbers, and must be factors of the least common multiple; hence, we divide every number that is divisible by either of these factors or by their 2 and 5; 12 by 2; 15 by 5; product; thus, we divide 10 by both and 75 by 5. We next divide the second line in like manner by 2 and 3; and afterwards the third line by 5. By this process we collect the factors of the given numbers into groups; and the product of the factors at the left of the vertical line is the least common multiple sought. 3. What is the least common multiple of 6, 15, 35, 42, and 70? OPERATION. 3,715 42 70 3 x 7 x 2 x 5 = 210, Ans. 70 must contain 6 and 35. SOLUTION. In this operation we omit the 6 and 35, because they are exactly contained in some of the other given numbers; thus, 6 is contained in 42, and 35 in 70; and whatever will contain 42 and Hence, we have only to find the least common multiple of the remaining numbers, 15, 42, and 70. RULE. -I. Write the numbers in a line, omitting any of the smaller numbers that are factors of the larger, and draw a vertical line at the left. II. Divide by any prime factor, or factors, that may be contained in one or more of the given numbers, and write the quotients and undivided numbers in a line underneath. III. In like manner divide the quotients and undivided numbers, and continue the process till all the factors of the given numbers have been transferred to the left of the verti cal line. Then find the product of these factors which will be the least common multiple required. |