5, it must divide the other factor 21, and the number may be resolved into the factors 5 × 3 × 7 or 15 x 7. Hence we deduce the following additional properties. 8. Every even number that is divisible by 3 is also divisible by 6; and every even number that is divisible by 9 is also divisible by 18. 9. Every number divisible by 3 or 9, in which the two terminating figures are divisible by 4, is divisible by 12 or 36. 10. Every number divisible by 3 or 9, whose terminating digit is 0 or 5, is divisible by 15 or 45. 11. Every prime number greater than 2, is one greater or one less than some multiple of 4. 12. Every prime number greater than 3, is one greater or one less than some multiple of 6. 13. Every number that has no prime factor, equal to, or less than its square root, is itself a prime number. For the product of any two factors, each greater than the square root of a number, would evidently be greater than the number itself. Therefore, if we attempt the division of any supposed prime, by all the primes less than its square root, and discover no factor, the number is itself a prime. TO FIND ALL THE DIVISORS OF A NUMBER. What numbers will divide 5940 without a remainder? 55 215940 We first resolve the number into all its prime factors, by 2,2970 commencing with 2 and dividing as often as possible, by 3 1485 each of the prime numbers in succession. We thus find 3 495 that 41580=22× 33× 5 × 11, or 2×2×3×3×3×5×11. 3 165 It may, therefore, have as many composite divisors as we can form distinct products of these prime factors. In order to determine all the possible products, we arrange 1, with the powers of the factor that is employed the greatest number of times, in a horizontal line. We then multiply each of the numbers in the first line, by each of the powers of another factor, each of the numbers of the preceding lines, by each of the powers of a third factor, &c., as in the following table. 11 11 1 The numbers of the first line having been arranged as directed, we multiply them separately by 2 and 22. All the numbers of these three lines, are multiplied by 5, which gives us three new lines of divisors. All the numbers of these six lines, are multiplied by 11, which gives us six new lines of divisors. We thus obtain 48 numbers that will divide 5940 without a remainder, and an examination of the table will show that these are all the divisors, since the prime factors are combined in every possible way. We are able to determine without actual trial, the number of exact divisors of any given number. By the foregoing table we perceive that 33 had 4, or 3 + 1 divisors. 33 × 28 has 12, or 3 + 1 × 2 + 1. 33 X 22 X 5 has 24 or 3 + 1 x 2 + 1 × 1 +- 1. In like manner each new factor can be multiplied by all the preceding divisors, as many times as are equivalent to the exponent of its power, thus forming so many new divisors, to be added to the preceding. Hence, for finding the number of divisors of any given number, we have the following RULE. Add 1 to the exponent of each of the prime factors of the given number, and multiply together the exponents thus increased. The product thus obtained, is the number of divisors sought. We have already seen that the greatest common divisor of two or more numbers, may be readily obtained by the aid of a table of prime factors. But by resolving fractions by inspection, into their prime factors, we may often reduce them to their lowest terms, without finding the greatest common divisor. For example, let it be required to reduce 68 105, 279, and 385 to their lowest terms. 85 132' 341' 10459 Resolving each fraction into its prime factors, we have 4 x 17 3 × 5 × 7 3 × 3 × 31 5 x 17 3 x 4 x 11' 5 X 7 × 11 and -" 11 × 31 5 x 11 x 19 Cancelling the factors common to the numerators and denominators, we have 4, 35 11 for the lowest terms of each fraction. 9 1. Resolve 65340 into its prime factors. 10813 10. How many integral divisors has 13600? 10943 22. Is 52099 a prime number? 23. How many integral divisors has 57660? 24. What are the prime factors of 168432? CHAPTER XXIII. NUMERICAL APPROXIMATIONS. THE student will have already perceived, in circulating decimals, and the extraction of surd roots, that there are many arithmetical operations which never give an exact result. There are also others, which, by a tedious process, would furnish an exact answer, but in which we desire only an approximate value, and we would gladly know what part of our labor may be omitted without affecting the accuracy required. A few of the most important NUMERICAL APPROXIMATIONS will form the subject of the present chapter. In ADDITION and SUBTRACTION of circulating decimals, it has been recommended to continue the repetends to five or six figures. If we wish to obtain the exact repetend, it will be necessary to change all the given repetends into others, containing as many figures as the least common multiple of the number of places in each repetend. Add 17.5, 3.7, 419.0875, 1.98563, and 32.1278. The numbers of repetend figures are 0, 1, 3, 2, and 4; the least common multiple of which is 12. The common repetend must therefore consist of 12 figures, commencing at the lowest place of the given repetends, which is ten-thousandths. The numbers will be written as follows. 17.500000000000000 999999999999 Adding the repetends, we find their sum 3.777777777777777 is 2814510279854. Dividing by the 419.087587587587587 rule for division by nines, we obtain a 1.985636363636363 quotient 2814510279856. The repetend 32.127812781278127 is written down, and the 2 carried to the 474.478814510279856 column of thousandths. Repetends, that thus commence and end at the same decimal places, are called similar and conterminuous. Subtract 1.742 from 2.937. 2.9379379 The common repetends have 6 figures. From 1.7424242 the right-hand figure of the remainder we subtract 1.1955136 1, because the repetend of the minuend is less than that of the subtrahend. The reason of this subtraction will become evident, if we change the repetends to fractions, and subtract 1.7424242 from 2.9379379 999999 999999 1. Add 17.69, 183, 25.75, 3.276, 194.43, and 649.287. 2. Add 3.219, 63.374, 285.12, 38.4, .0371, and 43.68. 3. Subtract 49.2871 from 64. 4. Subtract 215.9931 from 1842.2434. 5. Subtract 11.27 from 30.409. 6. Subtract 2856.036 from 3017.62591. 7. Subtract 43.763 from 288.1954. 8. Add 21.3, 28.72, 6.47, 19.345, 201.1593, and 419. 662434. 9. Add 7.83, 24.1, 79.142, 252.4163, and 17.3087. 10. Subtract 4.1956 from 21.28439113. In MULTIPLICATION, if only a certain degree of accuracy is desired, the product may be obtained by writing the units' figure of the multiplier under that figure of the multiplicand, whose place we would reserve in the product, and inverting the order of the remaining figures. In multiplying, we commence, for each partial product, with the figure of the multiplicand immediately above the multiplying figure, carrying the tens, which would arise from the multiplication of the two rejected figures at the right. Required the product of 287.613952 by 15.98421, correct to the fourth decimal place. 287.613952 12489.51 2876.1395 1438.0698 258.8525 23.0091 287.613952 15.98421 28 7613952 575 227904 11504 55808 1.1505 575 29 4597.2818 230091 1616 2588525 568 14380697 60 28761395 2 4597.28180769792 |