160.310941085484+471867 161. 473109731265÷590739 162. 178496100864÷6593184 163. 219614819645+4381075 164. 921073407384÷6310439 165. 813109865418÷8968578 166.930471076527+9481067.

Division is sometimes performed in which the figures of the products as they are found are subtracted, and the remainders only set down; and this is called Italian Division.

When the divisor is the product of two numbers, neither of which exceeds 12.

First divide by one of those numbers, and the quotient divide by the other.*

When there is not a remainder in the second division, that of the first, if any, is the true remainder: when there is not a remainder in the first division, that of the second multiplied by the first divisor, will give the true remainder; and if there is a remainder in each division, the second remainder multiplied by the first divisor, and added to the first remainder, will give the same result as if the operation had been performed by long division.

Eramples.

† Let 476859 be divided by 56, and 8510875 by 72.

. 19

131

590

148

475

43

* When the divisor consists of 3 or more parts, it is in general, easier and readier to employ long division.

In the above example the number 47 6859, is to be divided by 56, which is the product of 8 × 7 or of 7x8; first dividing by 7, the number contains 68122 sevens, and 5 over; these being divided by 8, gives a quotient of 8515, and has two left, which are two sevens ; therefore 2x7+5=19, the true remainder according to the rule, which is the same as is produced by the second operation of division.

If the divisor be an integer with a fraction annexed thereto, commonly called a mixed number.

Multiply the whole number by the denominator of the fraction, and to the product add the nnmerator for a new divisor, multiply also the given dividend by the same denominator, which will produce a new dividend; divide the new dividend by the new divisor, by any of the preceding methods which ever answers the purpose best, and the quotient is the answer.* Examples.

Let 439145 be divided by 34, and 1731415 by 31

4

15) 1756580

117105

4

3 3

By the substitution of another number in place of the given divisor, the operation of division may be often advantageously shortened, by first performing the division with the substituted number, as if it were the true one, and afterwards correcting the quotient by adding to, or subtracting therefrom, aecording as the substituted divisor exceeds, or falls short of the true one; thus, if our divisor was 999, and dividend 78834, by the substitution of 1000, we may at once perceive that the quotient is 78 with a remainder of 912, for by this substitution we have subtracted too many, as the dividend contained 1000, 78 times with a remainder of 834; and as 1000 exceeds 999 by 1, therefore by adding 78 to the first remainder, we obtain the true one: again, if our dividend was 364964, and divisor 1001, by a similar substitution of 1000, we at once perceive the quotient to be 364, with a remainder of 600, for here we have subtracted 364 too few, which being subtracted from the first remainder 964, leaves 600 for the true remainder required.

Some may object that the above more properly belongs to fractions, this I grant, yet there are n any who may not have learned, or who may never learn fractions, to whom it might be useful.

Here I have supposed the divisor to be only increased or diminished by one, but the same may be applied with equal advantage with a greater excess or deficiency; for example, let 74367 be divided by 993; here by the substitution of 1000 we may readily perceive the quotient to be 74, with a remainder of 885; for in the division of the given number by 1000, having subtracted 74 sevens too many, by the multiplication of 74 by 7, and the addition of the product to the former remainder 367, we obtain the true one: again, let 37642 be divided by 1009, we, with the same facility discover the quotient to be 37, with a remainder of 309; for here we have subtracted 9 times 37 too few, which being deducted from 642, the first remainder leaves 309 for the true one.*

When the divisor is a series of nines, division may be always performed with ease and certainty, let the dividend be ever so great, by dividing by an unit with as many ciphers annexed thereto as there are nines, and again dividing the quotient resulting by the same, and again, if necessary, until O be obtained for the integral quotient, preserving the several quotients and remainders; then adding each separately, and dividing the sum of the remainders, by the true divisor, and adding the quotient and remainder of this last division to the sum of the other quotients, the true quotient and remainder will be obtained.t

It will not be advantageous in general to employ this method, if the given divisor exceeds or falls short of that substituted by a greater excess or deficiency than 12, or if the given dividend is so great as to give a quotient the digits of which exceed those of the divisor.See this subject treated at large in John Walker's Philosophy of Arithmetic-pages 24 and 25.

+ Whenever the divisor is not more than 12 less than a number consisting of an unit with ciphers annexed, division may be performed with nearly the same facility; but here it will be necessary to multiply the several quotients by the deficiency, and add the several quotients and remainders.

If the divisor be a series of threes by multiplying the dividend by 3 and employing the above method, and dividing the remainder by 3, we obtain the same quotient and remainder as would result by dividing by the given divisor-If the divisor is a series of ones, multiply the dividend by 9, and divide as above directed, also divide the remainder thus found, by.nine, for the true remainder.

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