COMMON DIVISORS. 208. A Common Divisor of two or more numbers is a common factor of each of them. 209. The Greatest Common Divisor of two or more numbers is the greatest common factor, and is the product of all the common prime factors, and no more. 210. PRINCIPLES.-1. The only exact divisors of a number are its prime factors, or the product of two or more of them. 2. An exact divisor divides any number of times its dividend. 3. A common divisor of two or more numbers will divide their sum, and also the difference of any two of them. 4. The greatest common divisor of two or more numbers is the product of all their common prime factors. WRITTEN EXERCISES. 211. When the numbers can be readily factored. 1. What is the greatest common divisor of 42, 63, and 126? RULE.-I. Separate the numbers into their prime factors and find the product of all that are common. Or, I. Write the numbers in a line, and divide by any prime factor common to all the numbers. II. Divide the quotients in like manner, and so continue the division till all the quotients are prime to each other. III. The product of all the divisors will be the greatest common divisor. (PRIN. 4.) What is the greatest common divisor 8. Of 144 and 720 ? 9. Of 308 and 506? 10. Of 126, 210, and 252? 11. Of 72, 96, 126, and 384? 212. When the numbers cannot be readily factored. 1. Find the greatest common divisor of 507 and 1207. OPERATION. 1207 5272 1054 4593 153 682 136 684 17 ANALYSIS.--Draw two vertical lines, and place the larger number on the right, and the smaller on the left, one line lower down. Divide 1207 by 527, and write the quotient 2 between the vertical lines, the product, 1954, under the greater uumber, and the remainder 153, below. Next, divide 527 by this remainder 153, writing the quotient 3 between the verticals, the product 459, on the left, and the remainder C8, below. Again, divide the last divisor 153, by 68, and write the product and remainder in the same order as before. Finally, dividing the last divisor 68, by the last remainder 17, there is no remainder. Hence 17, the last divisor, is the greatest common divisor of 537 and 1207. In like manner find the greatest common divisor 2. Of 825 and 1372. |· 3. Of 1008 and 1036. RULE-I. Draw two vertical lines, and write the two numbers, one on each side, the greater number one line above the less. II. Divide the greater number by the less, writing the quotient between the verticals, the product under the dividend, and the remainder below. III. Divide the less number by the remainder, the last divisor by the last remainder, and so on, till nothing remains. The last divisor is the greatest common divisor sought. IV. If more than two numbers are given, first find the greatest common divisor of two of them, and then of this divisor and one of the remaining numbers, and so on to the last; the last common divisor found is the greatest common divisor of the given numbers. 4. What is the greatest number that will divide 2041 and 8476? 3281 and 10778 ? 5. What is the greatest number that will divide 216, 360, and 432? 141, 799, and 940? 6. B has $620, C $1116, and D $1488, with which they buy horses, at highest price per head that allows each man to invest all his money. How many horses does each buy? 7. A merchant has 15292 bushels of wheat, 1520 bushels of corn, and 504 bushels of beans, which he wishes to ship in the fewest bags of equal size that will exactly hold either kind of grain. How many bags will it take? MULTIPLES. 213. 1. What numbers between 5 and 30 are exactly divisible by 4? By 6? 7? 8? 9? 2. What numbers less than 40 are exactly divisible by 7? 3. What prime factors are common to 6, and 5 times 6? 4. Name some numbers exactly divisible by 4 and 6. 5. By what three prime numbers can 42 be divided ? 6. Name some numbers of which 3 and 4 are factors. 7. Find the least number exactly divisible by 3, 4, and 5. 214. A Multiple of a number is a number exactly divisible by the given number; or, it is any product or dividend of which the given number is a factor. Thus, 15, 20, and 25 are multiples of 5. 215. A Common Multiple of two or more given numbers is a number exactly divisible by each of them. 216. The Least Common Multiple of two or more given numbers is the least number exactly divisible by each of them. 217. PRINCIPLES.-1. A multiple of a number contains each of the prime factors of that number. 2. A common multiple of two or more numbers contains each of the prime factors of those numbers. Hence, 3. The least commor multiple of two or more numbers is the least number that contains each of the prime factors of those numbers. 4. A common multiple of two or more numbers may be found by multiplying the given numbers together. WRITTEN EXERCISES. 218. To find the least common multiple. 1. Find the least common multiple of 30, 42, and 66. OPERATION. 30= 2 × 3 × 5 422 x 3 x 7 66 = 2 × 3 × 11 2×3x11x7x5=2310 ANALYSIS.-The least common multiple cannot be less than the largest number 66, since it must contain 66; hence it must contain all the prime factors of 66, 2, 3, and 11. (PRIN. 1.) But the least common multiple of 66 must also contain all the prime factors of each of the other numbers, and since the prime factors 2 and 3 of 66 are common also to 42 and 30 omit them, and annex the factors 7 and 5 to those of 66, and the series 2, 3, 11, 7, and 5 are all the prime factors of the given numbers, and their product 2 × 3 × 11 × 7×5=2310, is the least common multiple of the given numbers. (PRIN. 3.) 2. Find the least common multiple of 24, 42, and 17. 3. Find the least common multiple of 8, 12, 20, and 30. 4. Find the least common multiple of 8, 12, 36, and 72. RULE.-I. Resolve each of the given numbers into its prime factors. II. Multiply together 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. Find the least common multiple 6. Of 52 and 78. 10. Of 12, 15 and 42. 11. Of 21, 35 and 56. |