Rule for performing Division. Place the divisor and dividend in the same line, separated by a small curved line; and on the right of the dividend draw another line of the same kind; inquire how often the first one or two figures on the left hand of the divisor are contained in the first one or more of those of the dividend, and place the result on the right as the first figure of the quotient: and the product arising from the multiplication of the divisor by this figure being subtracted from the dividend, bring down or annex to the remainder the next figure of the dividend, and let the same kind of operation be repeated till every figure of the dividend is disposed of; then the quotient, and the remainder, if any, will be ascertained. If the divisor do not exceed 12, these operations may be performed mentally, the quotient and remainder being placed in a line immediately under the dividend. 41. Since the quotient is the result arising from the division of the dividend by the divisor, it follows that the dividend must be the product arising from the multiplication of the divisor by the quotient, or of the quotient by the divisor also, if there be any remainder, it must evidently be added to this product to produce the true dividend, since the whole is equal to the sum of all its parts; and hence we have a method of shewing whether the division has been correctly performed, or not. Ex. Find the quotient and remainder when 275487 is divided by 736. The easiest proof of this operation, is that of adding together the figures of the remainder and the partial products of the divisor in vertical lines, since the sum thus formed ought manifestly to be equal to the dividend when the work is right, as in the following form which is omitted in practice. See the Appendix. 2208 5 1 5 2 2944 223 275487 the dividend. Abbreviations of Division. 42. The operation of Division may, in particular cases, be made to comprise fewer figures, or to take up less room, by such considerations as follow. Ex. 1. To divide 20573290 by 34500, we have where after the two ciphers in the divisor and the two figures 90 in the dividend are cut off, the operation is effected by the ordinary method, the said two figures of the dividend being annexed to the remainder at last, inasmuch as 112 from the places of the figures is equivalent to 11200. Ex. 2. Divide 792415 by 72. Here, since 72 is the product of 8 and 9, it is obvious from Ex. (2), of Article (35), that the quotient may be obtained from successive divisions by 8 and 9: 72 (8)792415 9) 99051, 7 first remainder : 1 1 0 0 5, 6 second remainder : and we have now only to deduce the true remainder from the two remainders just found. The dividend at first being so many units, the first remainder 7 must be units; but the second dividend being the result of the division by 8, must be regarded as so many times 8, and the second remainder will therefore be 6 times 8, or 48 units: whence, the true remainder will be and we may lay down a rule in the following words. In dividing by two numbers, instead of one equal to their product, the true remainder is equal to the product of the last remainder and the first divisor, together with the first remainder. 43. Examples for Practice in Division. (1) 2) 348 (2) 3) 4 596 (3) 4) 27,628 4 (4) 5) 84375 (5) 6) 5 3 8 44 (6) 7) 536074 and prove (19) Find the quotient of 76294 by 32: of 729518 by 49: of 8015473 by 66, and of 4050873 by 121; the correctness of the operations. 44. The operation of Division is expressed by means of the sign and sometimes, which is read by, or divided by; thus, 42+7=6 implies that the result of the division of 42 by 7 is 6: again, (70-7)+(4+5) is equivalent to 63÷ 9=7. MEASURES AND MULTIPLES. 45. DEF. 1. A Measure of a number is any number which divides it without a remainder; as 4 is a measure of 24, because it is contained exactly 6 times in 24. It is said to measure the number by the units contained in the quotient. All numbers have 1 for a measure; those, whereof 2 is a measure, are called even numbers, admitting of being divided into two equal parts; and all other numbers are termed odd numbers. 46. DEF. 2. A Common Measure of two or more numbers is any number, which will divide each of them without leaving a remainder; and the greatest of such measures is called the Greatest Common Measure, or Greatest Common Divisor: thus, 3 is a common measure of 18 and 30; whereas 6 is their greatest common measure, being the greatest number capable of dividing each of them without a remainder. 47. DEF. 3. An Aliquot Part of a number is any measure of it. 48. DEF. 4. A Multiple of a number is any number which is divisible by it, or contains it an exact number of times; as 108 is a multiple of 12, because 12 is contained exactly 9 times in 108. 49. DEF. 5. A Common Multiple of two or more numbers is any number which is divisible by each of them separately; and the Least Common Multiple is the least number that can be divided by each of them without a remainder as 24 is a common multiple of 3 and 4, because divisible by both of them; whereas 12 is their least common multiple, because it is the least number that both 3 and 4 can divide without leaving a remainder. 50. DEF. 6. A Composite Number is one which arises from the multiplication of two or more other numbers termed Factors; and it is thus distinguished from an Incomposite or Prime Number, which cannot so originate: as 22 is a composite number, because it is equal to the product of the factors 2 and 11; but 11 is an incomposite or prime number, because the multiplication of no two or more factors will produce it, unity, which is merely the element of number, being excepted. 51. If one number measure each of two others, it will measure their sum and difference: also, any multiples of each, their sums and differences. Thus, 4 is a common measure of 20 and 12; and a multiple of 12 = 12 × 7 = 84 = 4 × 21 : each of which evidently comprises the number 4 as a measure or factor: and similarly of more numbers. 52. To find the greatest common measure of two numbers. Let the numbers proposed be 63 and 168: then resolving each of them into its prime factors, we have 63=7x9=7× 3 × 3: 1687 x 247 × 3 × 8=7×3×2×2×2: and the greatest common measure is evidently 7 x 3 or 21, because 3 and 2 × 2 × 2 or 8 have no common factor : and employing the principles of the last Article, we obtain the same result by the following form: 63) 168 (2 42)63 (1 21) 42(2 4 2 where 21 the last Divisor is the greatest common measure: and we have hence the following Rule. Rule for finding the Greatest Common Measure. Divide the greater of the numbers by the less, and then the divisor by the remainder repeat this opera |