LEAST COMMON MULTIPLE. 114. A MULTIPLE of a number is any product of which the number is a factor; hence, any multiple of a number is exactly divisible by the number itself. 115. A COMMON MULTIPLE of two or more numbers is any number which each will divide without a remainder. 116. THE LEAST COMMON MULTIPLE of two or more numbers is the least number which they will separately divide without a remainder. 117. Principles-Operations-and Rule. 1. Any divisible number, is divisible by any prime factor of the exact divisor. 2. If a number has several exact divisors, it will be divisible by all their prime factors. 3. Hence, the question of finding the least common multiple of sev eral numbers is reduced to finding a number which shall contain all their prime factors, and none others. 1. What is the least common multiple of 6, 12, and 18? OPERATION. 2)6 12 3)3 1 6 2 ..18 9 3 ANALYSIS.-Having placed the given numbers in a line, if we divide by 2, we find the quotients 3, 6 and 9; hence, 2 is a prime factor of all the numbers. Dividing by 3, we find that 3 is a prime factor of the quotients 3, 6, and 9; and hence, the quotients 2 and 3 are prime factors of 12 and 18; therefore, the prime factors of all the numbers are 2, 3, 2 and 3; and their product, 36, is the least common multiple. 2 3 × 2 × 3 = 36 114. What is a multiple of a number?-115. What is a common multiple of two or more numbers?-116. What is the least common multiple of two or more numbers? 117. What is the first principle on which the operation for finding the least common multiple depends? What is the second? What is the third? Give the rule for finding the least common multiple. Rule. I. Place the numbers on the same line, and divide by any PRIME number that will exactly divide two or more of them, and set down, in a line below, the quotients and the undivided numbers: II. Then divide as before, until there is no prime number greater than 1 that will exactly divide any two of them: III. Then multiply together the divisors and the numbers of the lower line, and their product will be the least common multiple. NOTE. If the numbers have no common prime factor, their product will be their least common multiple. Examples. 1. What is the least common multiple of 4, 9, 10, 15, 18, 20, 21? 2. What is the least common multiple of 8, 9, 10, 12, 25, 32, 75, 80? 3. What is the least common multiple of 1, 2, 3, 4, 5, 6, 7,9? 4. What is the least common multiple of 9, 16, 42, 63, 21, 14, 72? 5. What is the least common multiple of 7, 15, 21, 28, 35, 100, 125? 6. What is the least common multiple of 15, 16, 18, 20, 24, 25, 27, 30? 7. What is the least common multiple of 9, 18, 27, 36, 45, 54? 8. What is the least common multiple of 4, 10, 14, 15, 21? 9. What is the least common multiple of 7, 14, 16, 21, 24? 10. What is the least common multiple of 49, 14, 84, 168, 98? 11. A can dig 9 rods of ditch in a day; B, 12 rods in a day; and C, 16 rods in a day: what is the smallest number of rods that would afford exact days of labor to each, working alone? In what time would each do the whole work? 12. A blacksmith employed 4 classes of workmen, at $15, $16, $21, and $24 per month, for each man respectively, paying to each class the same amount of wages. Required the least amount that will pay either class for 1 month; also, the number of men in each class? 13. A farmer has a number of bags containing 2 bushels each; of barrels, containing 3 bushels each; of boxes, containing 7 bushels each; and of hogsheads, containing 15 bushels each : what is the smallest quantity of wheat that would fill each an exact number of times, and how many times would that quantity fill each? 14. Four persons start from the same point to travel round a circuit of 300 miles in circumference. A goes 15 miles a day; B, 20 miles; C, 25 miles; and D, 30 miles a day. How many days must they travel before they will all come together again at the same point, and how many times will each have gone round? NOTE.-First find the number of days that it will take each to travel round the circuit. GREATEST COMMON DIVISOR. 118. A COMMON DIVISOR of two or more numbers, is any number that will divide each of them without a remainder; hence, it is always a common factor of the numbers. 119. THE GREATEST COMMON DIVISOR of two or more numbers, is the greatest number that will divide each of them without a remainder; hence, it is their greatest common factor. 120. Two numbers are said to be prime to each other, when they have no common divisor. NOTE.-Since 1 will divide every number, it is not reckoned among the common divisors. 118. What is a common divisor?-119. What is the greatest com. mon divisor? 120. When are two numbers said to be prime to each other? 121. To find the greatest common divisor of two or more numbers, when the numbers are small. Since an exact divisor is a factor, the greatest common divisor of the given numbers will be their greatest common factor: hence, I. Resolve each number into its prime factors, and observe those which are common to all the numbers: II. Multiply the common factors together, and their product will be the greatest common divisor. Examples. 1. What is the greatest common divisor of 12 and 20? ANALYSIS.-There are three prime factors in 12; viz., 2, 2, and 3: there are three prime factors in 20; viz., 2, 2, and 5. The factors 2 and 2 are common; hence, 2 x 2 = 4 is the greatest common divisor. OPERATION. 12 = 2 x 2 x 3. 20 = 2 × 2 × 5. 2. What is the greatest common divisor of 18 and 36? 3. What is the greatest common divisor of 12, 24, and 60? 4. What is the greatest common divisor of 15, 50, and 40? 5. What is the greatest common divisor of 24, 18, and 144? 6. What is the greatest common divisor of 50, 100, and 80? 7. What is the greatest common divisor of 56, 84, and 140? 8. What is the greatest common divisor of 84, 154, and 210? 122. To find the greatest common divisor, when the numbers are large. This method depends on the following principles: 1. Any number which will exactly divide two numbers separately, will divide their difference; else, we should have a whole number equal to a fraction, which is impossible. ILLUSTRATION. 30-8=22 121. How do you find the greatest common divisor, when the numbers are small?-122. On what principle does finding the greatest common divisor depend? Give the rule. 2. Any number that will exactly divide the difference of two numbers, and one of them, will exactly divide the other: else, we should have a whole number equal to a fraction, which is impossible. 3. Any number which will exactly divide another, will divide any multiple of that other; because, the first dividend which is divisible, is a factor of the multiple. 1. Let it be required to find the greatest common divisor of the numbers 216 and 408. ANALYSIS.-The greatest common divisor cannot be greater than the least number, 216. Now, as 216 will divide itself, let us see if it will divide 408; for, if it will, it is the greatest common divisor. Making the division, we find a quotient 1, and a remainder, 192; hence, 216 is not a common divisor. OPERATION. 216) 408 (1 192) 216 (1 192 24) 192 (8 192 The greatest common divisor of 216 and 408 will divide the remainder 192; and if 192 will exactly divide 216, it will be the greatest common divisor. We find that 192 is contained in 216 once, and a remainder 24. The greatest common divisor of 192 and 216 will divide the remainder 24; and if 24 will exactly divide 192, it will also divide 216, and consequently 408; now, 24 exactly divides 192, and hence is the greatest common divisor sought. Rule. Divide the greater number by the less, and then divide the preceding divisor by the remainder, and so on, till nothing remains: the last divisor will be the greatest common divisor. Principles from the Rule. 1. If the last remainder is 1, the numbers are prime to each other. 2. If, in the course of the operation, any one of the remainders is a prime number, and will not exactly divide the preceding divisor, it is certain that there is no common divisor. 3. To find the greatest common divisor of three or more numbers, find the greatest common divisor of two of them, and then the divisor of this common divisor, and of the third number, and so on. |