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 × 22 has 12, or 3 + 1 × 2 + 1. 33 X 22 X 5 has 24 or 3 + 1 × 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 10459 §8, 132, 68 105 279, and 385 to their lowest terms. Resolving each fraction into its prime factors, we have 9 7 Cancelling the factors common to the numerators and de- 1. Resolve 65340 into its prime factors. 10813 9. Reduce 542 to its lowest terms. 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 com. mence, 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. The units' figure of the multiplier being placed under the 4th decimal of the multiplicand, and the whole multiplier reversed, the product of each figure by the one above it will be ten-thousandths. Therefore the right-hand figure of each partial product, will fall in the column of ten-thousandths. In the second product, multiplying 52 by 5 we obtain 260, which being nearer 300 than 200, we carry 3 to the product of 9 by 5. The multiplication has also been performed in the usual way, the vertical line showing the figures that are rejected. If the multiplicand does not contain enough decimal figures to correspond with the inverted multiplier, the deficiency should be supplied by annexing zeroes. The same contrac tion may be applied to integers, if we wish only to obtain the thousands, millions, &c., of the product. 11. Required the product of 2869.174381 by 154.49216, true to three places of decimals. 12. Find the product of 176.2428 by 119.43, true to the second decimal place. 13. What is the integral part of the product of 49821.476 by 25.341 ? 14. What is the integral part of the product of 51763. 84926 by 2.4957 ? 15. Multiply 778148.3219 by 954.638, and reserve two decimal places. 16. Multiply 11817.93642 by 2581.36, and reserve two decimal places. 17. Multiply 4435.81977 by 6.9043, and reserve one decimal place. 18. What is the product of 7716.4295 by 19.87436, within .001 ? 19. Find the product of 63917.48219 by 587.618, within one ten-thousandth. 20. Find the product, true to the 4th decimal place, of 21.87964 by 2.38917. In DIVISION, a similar contraction may be made when the the divisor is large, which is also applicable in the extraction of roots. The first quotient figure is of the same numerical value as |