ANOTHER FORM.-In the margin on the right is another form of writing the division, which in practice we prefer to the above. The problem is to find the greatest common divisor of 32 and 116. The method will be clear from a slight inspection of the work. Let the pupils adopt it after they are familiar with the common form. OPERATION. 32 116 3 96 20 201 12 121 8 81 82 Rule.-Divide the greater number by the less, the divisor by the remainder, and thus continue to divide the last divisor· by the last remainder until there is no remainder; the last divisor will be the greatest common divisor. NOTE. To find the greatest common divisor of more than two numbers, we first find the greatest common divisor of two of them, then of that divisor and one of the other numbers, etc. EXAMPLES FOR PRACTICE. Find the greatest common divisor of 10. A farmer has two heaps of apples, one containing 364, and the other 585, which he wishes to divide into smaller heaps, each containing the same number; what is the largest number that the heaps may contain? Ans. 13. 11. A benevolent society distributed $678, $906, and $1146 in equal sums to the poor of three wards of a city, the sums being as large as possible. Required the amount of the equal sums and the number of persons receiving relief in each ward. Ans. $6; 113; 151; 191. 12. A Western landholder has three tracts, the first containing 533 acres, the second 574 acres, and the third 861 acres, which he wishes to divide into fields of equal size, having the least number possible. Required the number of fields, and the number of acres in each. Ans. 48 fields; 41 acres. INTRODUCTION TO COMMON MULTIPLE. MENTAL EXERCISES. 1. What number is three times 5? four times 6? five times o? six times 8? 2. A number which is one or more times another number is called a multiple of that number. 3. What is the multiple of 4? of 5? of 6? of 7? of 8? of 9? of 10? of 11? of 12? 4. What multiple is common to 2 and 3? to 3 and 4? to 4 and 6? to 6 and 8? to 6 and 9? 5. What may we call a multiple common to two or more numbers? Ans. A common multiple. 6. What is a common multiple of 4 and 5? 8 and 9? 6 and 7? 4 and 6? 5 and 8? 7. What is the least multiple common to 2 and 4? 4 and 6? 4 and 8? 6 and 8? 8 and 12? 8. What shall we call the least multiple common to two or more numbers? Ans. Their least common multiple. 9. What is the least common multiple of 4 and 6? 9 and 12? 10 and 16? 20 and 24? 25 and 30? 10. What is the least common multiple of 6 and 8? 8 and 12? 12 and 16? 16 and 32? 35 and 70? LEAST COMMON MULTIPLE. 124. A Multiple of a number is one or more times the number; thus, 4 times 5, or 20, is a multiple of 5. 125. A Common Multiple of two or more numbers is a number which is a multiple of each of them; thus, 24 is a common multiple of 2, 3, and 4. 126. The Least Common Multiple of two or more numbers is the least number which is a multiple of each of them; thus, 12 is the least common multiple of 2, 3, and 4. NOTE. The least common multiple may be represented by the initials L. C. M. PRINCIPLES. 1. A multiple of a number is exactly divisible by that number. 2. A multiple of a number must contain all the prim factors of that number. 3. A common multiple of two or more numbers must contain all the prime factors of each of those numbers. 4. The least common multiple of two or more numbers must contain all the prime factors of each number, and no other factors. CASE I. 127. When the numbers are small and easily factored. FIRST METHOD. 128. This method consists in resolving the numbers into their prime factors, and taking the product of all the different factors. 1. Find the least common multiple of 12, 30, and 70. OPERATION. 12=2×2×3 70=2X5X7 2×2×3×5×7=420 SOLUTION. We first resolve the numbers into their prime factors. A multiple of 12 must contain the factors of 12, 2, 2, 3; a multiple of 30 must contain the factors of 30, 2, 3, 5; a multiple of 70 must contain the factors of 70, 2, 5, 7; hence the common multiple of 12, 30, and 70 must contain all these different factors and no others; therefore 2×2×5×3×7, or 420, is the L. C. M. of 12, 30, and 70 (Prin. 4). Rule.-I. Resolve the numbers into their prime factors. II. Take the product of all the different factors, using each factor the greatest number of times it occurs in either number. NOTE. Any numbers which are divisors of the others may be omitted, since the multiple of the other numbers will be a multiple of these. 10. A has $14, B $15, C $36, and D as many as the least common multiple of the amounts of the others; how many has D? Ans. $1260 SECOND METHOD. 129. This method consists in taking out the prime fac tors of the least common multiple and finding their product 1. Find the least common multiple of 12, 30, and 70. OPERATION. 212-30 - 70 3 6 15 35 5 2 2 5 35 1 7 SOLUTION.-Placing the numbers one beside another and dividing by 2, we see that 2 is a factor of each of them, it is therefore a factor of the L. C. M. (Prin. 3); dividing the quotients that will contain it by 3, we see that 3 is a factor of some of the numbers, it is therefore a factor of the L. C. M. Dividing the next quotients by 5, we see that 5 is a factor of some of them, hence 5 is a factor of the L. C. M.; and the quotients having no other common factor, we see that the factors of the given numbers are 2, 3, 5, 2, and 7, hence their product, which is 420, is the L. C. M. Hence the following 2×3×5×2×7=420 Rule.-I. Write the numbers one beside another, divide by any prime number that will exactly divide two or more, and write the quotients and undivided numbers beneath. II. Divide the quotients in the same manner, and thus continue until no two numbers in the lowest line have a common factor. III. Take the product of the divisors and final quotients; the result will be the least common multiple required. Find the least common multiple of 2. 16, 20, and 30. 3. 28, 56, and 84. 4. 48, 60, and 30. 5. 150, 200, and 250. 6. 40, 96, 100, and 120. 7. 120, 180, 200, and 240. 8, 140, 280, 160, and 320. Ans. 240. Ans. 168. Ans. 240. Ans. 3000. Ans. 2400. Ans. 3600. Ans. 2240. CASE II. 130. When the numbers are large and cannot be readily factored. OPERATION. 28=4x7; 63=9x7 L. C. M. 1. Find the least common multiple of 28 and 63. SOLUTION. The greatest common divisor of these numbers is 7; 28 equals 4 times 7, and 63 equals 9 times 7; hence the L. C. M., as found in the first method, is 4 X 7 X 9, which equals 28 multiplied by 63 divided by 7; or the first number multiplied by the second divided by their greatest common divisor. 4X7X9 63 28 X 7 Rule.-I. Find the greatest common divisor of two num bers, divide one number by it, and multiply the other number by the quotient. II. When there are more than two numbers, find the least common multiple of two of the numbers, and then of this number and the third number, etc. 9. What is the smallest sum of money for which I could hire workmen for one month, paying either $16, $20, $28, or $35 a month? Ans. $560 10. A can dig 14 rods of ditch in a week, B 18 rods, C 22 rods, and D 24 rods; what is the least number of rods that would afford an exact number of weeks' work for each one of them? Ans. 5544 rods. 11. What is the smallest number of bushels of corn that will fill a number of barrels containing 3 bushels each, a number of sacks containing 5 bushels each, a number of casks containing 14 bushels each, or a number of bins containing 48 bushels each? Ans. 1680. 12. Four men start at the same place to walk around a garden; A can go around in 9 minutes, B in 10 minutes, C in 12 minutes, and D in 15 minutes; in what time will they all meet at the starting point? Ans. 180 minutes. 13. A, B, C, and I start from the same point, A traveling a mile in 18 minutes, B in 24 minutes, C in 30 minutes, and D in 35 minutes; what is the least whole number of miles each may travel that they may return to the starting point at the same moment? Ans. A, 140; B, 105; C, 84; D, 72. |