24. Divide 22320 by 240. 2408 × 6 × 54 X 12 X 5 4 X 6 X 10= etc. 57. Should the learner find a difficulty in determining the remainder, he has but to remember that it is always of the same kind as the dividend (Art. 50). 27. Divide 86 by 21. OPERATION. 7)86 3) 12...2 Rem. Quotient, 4 28. Divide 92 by 28. OPERATION. 4)92 7)23 Quotient, 3...2 Rem. 29. Divide 527 by 42. OPERATION. 6) 527 7) 87...5 Rem. Quotient, 1 2...3 Rem. In this example, as 86 is the true dividend, 2 is the true remainder. In this example, as 23 is only one fourth of the true dividend, so the remainder, 2, is only one fourth of the true remainder, .. 2 X 48, true remainder. By the explanations of the last two examples, we see that 5 is one part of the true remainder, and that 3, the second remainder, multiplied by 6, the first divisor, is the other part; i. e. 5 +3 × 6 - 23 true remainder. The same species of reasoning applies when there are more than two divisors. Hence, To obtain the true remainder when division is performed by using the component parts of the divisor, RULE 1.- Multiply each remainder except that left by the first division, by the continued product of the divisors preceding that which gave the remainders severally, and the sum of the products together with the remainder left by the first division, will be the true remainder. RULE 2. · Multiply the last remainder by the divisor preceding that which gave the last remainder and to the product add the preceding remainder; multiply this sum by the preceding divisor and add the preceding remainder, and so proceed until the first remainder is added; the sum so obtained will be the true remainder. Should any remainder be 0, then 0 is to be added. APPLICATION OF THIS RULE TO Ex. 30. 7×5+4= 39; 39 × 7+ 3 = 276, true remainder, as by Rule 1. 31. Divide 5273 by 42. 42 2 × 3 × 7. Ans. 125 and 23 Rem. 32. Divide 46987 by 504, using the factors of the divisor. 58. To divide by 10, cut off, by a point, one figure from the right of the dividend; the figures at the left of the point are the quotient, and that at the right is the remainder. The reason is obvious. By taking away the right-hand figure, each of the other figures is brought one place nearer to units, and its value is only one tenth as great as before (Art. 16), .. the whole is divided by 10. 38. Divide 756 by 10. Ans. 75.6, i. e. 75 Quo. and 6 Rem. 39. Divide 402763 by 10. (a) For like reasons we cut off two figures to divide by 100, three to divide by 1000, and, generally,* we cut off as many figures from the right of the dividend as there are ciphers in the divisor. 40. Divide 76943 by 100. 41. Divide 98765423 by 100000. Ans. 769 and 43 Rem. Ans. 987 and 65423 Rem. 42. Divide 3078654321 by 100000000. (b) To divide by 20, 50, 700, 56000, or any similar number, cut off as many figures from the right of the dividend as there are ciphers at the right of the significant figures of the divisor, and then divide the remaining figures of the dividend by the significant figures of the divisor. This is on the principle of dividing by the component parts of the divisor, .. the true remainder will be found by the rules in Art. 57. 43. Divide 74689 by 8000. OPERATION. 8) 74.689 Ans. 9 and 2689 Rem. We divide by 1000 by cutting off 689, which gives 74 for a quotient and 689 for a remainder; then divide 74 by 8 and obtain the quotient, 9, and remainder, 2. This remainder, 2, is 2000, which, increased by 689, gives 2689 for the true remainder (Art. 57, Rule 2). Quotient, 9...2 Rem. * Generally, in mathematics, means universally. 44. Divide 67475 by 2400. 45. Divide 74689 by 4200. 46. Divide 276987 by 3300. Ans. 17 and 3289 Rem. Ans. 17 and 4842 Rem. 53. A certain product is 1728, and one of the factors is 12; what is the other factor? Ans. 144. 54. How many times is 157 contained in 74732 ? 55. By what must 316128 be divided to give 356 for a quotient? 56. By what must 87 be multiplied to produce 83868? 57. If 1357901 be a dividend and 87 the divisor, what is the quotient? remainder? ? 58. Dividend 6789468; quotient =1234; divisor = 59. A dividend is 6481, the quotient is 72 and the remainder is 73; what is the divisor? 60. Dividend = 98765; divisor 17; remainder =? 61. The product of three numbers is 16777216, and the product of two of them is 131072; what is the other number? 62. 248832 = 144 X ? 59. The value of a quotient depends upon the relative values of the divisor and dividend and not upon their absolute values, as will be seen by the following propositions. (a) If the divisor remains unaltered, multiplying the dividend by any number is, in effect, multiplying the quotient by the same number; thus, 1535 10043 i.e. multiplying the dividend by 4 multiplies the quotient by 4. (b) Dividing the dividend by any number is dividing the quotient by the same number; thus, i. e. dividing the dividend by 3 divides the quotient by 3. (c) Multiplying the divisor divides the quotient; thus, i. e. multiplying the divisor by 3 divides the quotient by 3. (d) Dividing the divisor multiplies the quotient; thus, i. e. dividing the divisor by 5 multiplies the quotient by 5. dend the greater the quotient, and the reverse. (f) COROLLARY TO (c) AND (d). — The greater the divisor the less the quotient, and the reverse. 60. If a number be multiplied by any number, and the product be divided by the multiplier, the quotient will be the multiplicand; thus, 8 X 756, and 56-78, the multiplicand. COROLLARY. Since multiplying the dividend multiplies the. quotient (59, a), and multiplying the divisor divides the quotient (59, c), .. multiplying both dividend and divisor by the same number does not affect the quotient; thus, 123 2464, Quotient unchanged. * A corollary is an inference from preceding reasoning. Is divides 2: : 12 |