Example.
Divide 87654213, by 937. 987)87654213(88808 Quotient.
7896 .8694 7896 .7982 .7896 .8613 .7896 .717 remainder.
88808 987 621663 710465 799279 87654213 proof.
Ans. 88808ii+ To prove the truth of the sum, I mut. tiply the quotient by the divisor, and take in the remainder, which gives the nal dividend.
Examples for Practice.
1. Divide 721354 by 2. . . 57214372 by 42 3. . . 67215731 by 63 4. . . 802594321 by 84 5. . . 965314162 by 89 6. . . 43219875 by 674 7. . . 57397296 by 714 8. . . 496521 by 2798 9. . . 49446327 by 796 10. . . 47314967 by 699 11. . . 275472734 by 497 12. . . 43927483 by 586 13. . . 96543245 by 763 14. . . 25769782 by 469 A number that divides another without a remainder is said to measure it, and the several numbers that measure another are called its aliquot parts. Thus 3, 6, 9, 12, 18, are the aliquot parts of 36. As it is frequently necessary to discover num bers which measure others, if may be ob served, 1. That every number ending with an even number, that is, with 2, 4, 6, 8, or 0, is measured by 2. 2. Every number, ending with 5 or 0, is measured by 5. 3. Every number, whose figures, when added, amount to an even number of 3's or 9's, is measured by 3 or 9 re spectively.
In speaking of the contractions and va riety in division, we have already seen, that when the divisor does not exceed 12, the whole computation may be perform ed without setting down any figure ex cept the quotient.
When the divisor is a composite num ber, we may divide successively by the component parts : thus, if 678450 is to be divided by 75, we may either perform the operation by long division, or divide by 5, 5, and 3, because 5 x 5 x 3=75.
Where there are cyphers annexed to the divisor, cut them off, and cut off also an equal number of figures from the di vidend; annex these figures to the re mainder.
Example. Divide 54234564 by 602400.
6024,00)542345,64 54216 ...18564 To divide by 10, 100, 1000, &c. Cut off as many figures on the right hand of the dividend as there are cyphers in the di visor. The figures which remain on the left hand compose the quotient, and those cut off compose the remainder.
Example.
Divide 594256 by 1000. 1,000)594,256 Ans. When the divisor consists of several figures, we may try them separately, " by enquiring how often the first figure of the divisor is contained in the first figure of the dividend, and then considering whe ther the second and following figures of the divisor be contained as often in the corresponding ones of the dividend with the remainder, if any prefixed. If not,
we must begin again, and make trial of a lower number.
We may form a table of the products of the divisor multiplied by the nine di gits, in order to discover more readily bow often it is contained in each part of the dividend. This is always useful, when the dividend is very long, or when it is required to divide frequently by the same divisor.
Example, Divide 689543271 by 37. 37 x 2=-- 74 37)689543271(18636304 3=111 37 4=-148 319 5 =-185 226 6=222 235 7=259 222 8=296 9=333 , 134 111 .233 222 . 112 111 • 171 148 . 23 As multiplication supplies the place of frequent additions, and division of fre quent subtraction, they are only repeti tions and contractions of the simple rules, and when compared together, their ten dency is exactly opposite. As numbers increased by addition are diminished and brought back to their original quantity by subtraction, in the same manner numbers compounded by multiplication are reduc ed by division to the parts from which they are compounded. The multiplier shows how many additions are necessary to pro duce the number, and the quotient shows how many subtractions are necessary to exhaust it. Hence it follows, that the pro duct divided by the multiplicand will give the multiplier ; and because either factor may be assumed for the multiplicand., therefore the product divided by either factor gives the other. It also follows, that the dividend is equal to the product of the divisor and quotient multiplied tow gether, and of course these operations mutually prove each other.
To prove Multiplication. Divide the pro duct by either factor; if the operation be right, the quotient is the other factor, and there is no remainder.
To prove Division. Multiply the divisor and quotient together ; to the product add the remainder, if any ; and if the operation be right, it makes up the divi dend.—We proceed to