2. Divide 147150 by 45. 45+of itself = 50. 5.0)14715.0 9)2943 quotient by 50. 327of itself added. 3270, Ans. A remainder produced by the first divisor, is so many parts of the true divisor, and the second remainder is such a part of the true remainder as the second divisor is of the true divisor. 3. Divide 78360 by 75. 75+of itself = 100. 3)783.60 quotient by 100. 261 of the whole number added. 1044, Ans. == Here, the remainder found by dividing by 100, is 60, which, as will be seen, is written in the quotient as so many seventy-fifths instead of hundredths. This is the true result; for the remainder, as well as the whole number, must be increased by of itself; and when so increased, it will be equal to as many seventy-fifths as it was hundredths: + 2 which reduced $. The rule will hold good in all cases; the required increase will make the first remainder the numerator of a fraction whose denominator is the true divisor. Therefore it may be taken as such without change. = But, by dividing the decimal also by 3, and adding, we shall obtain the true quotient, with a decimal remainder. Thus, 3)783.60 1044.80, true quotient, which is the same in value as 104498, as before shown. 4. Divide 278917 by 75. 3)2789.17, quotient by 100; 1st rem. 17 929.2, added; 2d rem. 2. 2 × 25 = 50 371843 True remainder, 67 75 ÷ 3 = 25, and hence 2, the second remainder, is of the true amount. 2 x 25+ 17 = 67, is the true remainder. Or, continuing the division by 3, we have 891 100 3)2789.17 3718.891, True quotient. reduced will be found to be equal to, as before. In the latter solution, the remainder is greater than the divisor. §4 = 138, which added to 208 gives the true quotient. 49. To divide by any number under and near 100, 1000, etc. RULE.-I. Divide by a unit of the order next higher than the divisor, by cutting off figures from the right of the dividend. II. Multiply the whole number of the quotient so obtained by the complement of the divisor, cut off the same number of figures as before, and write the result under the first quotient. III. Proceed in like manner with the whole number of this quotient, and so on till no whole number remains to multiply. IV. Add the several results together; the sum will be the quotient (the whole number being on the left of the point, and the remainder on the right), subject to the following changes, when required: V. If any number has been carried from the remainder in adding, multiply this number also by the complement of the divisor, point off as above, and add the result to the quotient. VI. If the remainder equal or exceed the divisor, reduce it to a whole or mixed number, and add the same to the quotient. EXAMPLES. EXPLANATION.-35714 ÷ 100 = 367 and 14 remainder. Now, as 100 99 + 1,99 is contained in the same number 367 times, with a remainder of 367 × 1 + 14; or the quotient 367 + 14 may be written, 367 + 99 = 1. Divide 36714 by 99. 367.14, quotient by 100. 3.67, 367 × 1÷100. 3, 3 × 1 100. 370, Ans. 367 + 357. Therefore, 3673 is taken as a partial quotient, and 367 remains to be divided. Proceeding as before, 367 ÷ 100 3.67; hence, 367 ÷ 99 = 3, with 3+ 67 remainder. Therefore, 3 is another part of the true quotient, while 3, divided in the same manner, gives for the final part of the remainder. Adding these partial quotients, we have 37033, the true quotient. 2. Divide 1673458 by 996. 1673.458, quotient by 1000. 6.692, 16.73 × 4 ÷ 1000. 6 x 4 1000. 24, 4, 1 (to be carried) = EXPLANATION.-Reasoning as in the last case, we continually multiply the whole number of the quotient obtained by dividing by 1000, by 4, the complement of the divisor, and place the products under the first. After the third division, no whole number remains; but, we observe that, in adding, 1 must be carried from the sum of the partial remainders; therefore, we multiply this 1 by 4, divide it by 1000, and place the quotient under the other remainders. The whole quotient is then found to be 16809%. 168078 3. Divide 64578 by 98. 6. Divide 87432567 by 997. DIVISORS, FACTORS, AND MULTIPLES. 50. An Exact Divisor, or Measure of a number, is any whole number that will divide it without a remainder. Thus, 2, 3, 4, and 6 are exact divisors of 12. 51. The Factors of a number are the whole numbers which multiplied together will produce it. Thus, 3 and 4 are factors of 12. 52. Every factor of a number must be a divisor of it; hence, divisors and factors are convertible terms; a divisor is a factor, and vice versa. 53. A Multiple of a number is any number which is exactly divisible by it. Thus, 6, 9, and 12 are multiples of 3. 54. A Prime Number is a number which has no exact divisor, except itself, and 1; that is, which cannot be resolved into two or more integral factors; as 2, 5, 7, 9, 11, 13, etc. 55. A Composite Number is a number which has other exact divisors besides itself and 1; as 4, 8, 10, 12. 56. The Prime Factors of a number are the prime numbers which multiplied together produce that number. Thus, 2, 3, and 5 are the prime factors of 30. 57. A Composite Factor of a number is a factor which is a composite number. Thus, 15 is a composite factor of 30. METHOD OF FACTORING NUMBERS. 58. A number is said to be resolved into its prime factors when it is separated into factors which are all prime numbers. 59. A factor is a divisor; therefore, it is obvious that the prime factors of a number are all of its exact prime divisors. 60. To find the prime factors of a composite number. EXAMPLE. What are the prime factors of 15015? OPERATION. 5)15015 3)3003 7)1001 ANALYSIS.-By dividing by any exact prime divisor, we separate the given number into two factors, one of which at least (the divisor) is prime. If the quotient be also prime, it is the only other prime factor, and the work is ended; if not, it contains some exact divisor which is another of the prime factors of the original number. So, by continuing the division till a prime quotient is obtained, all the divisors and the quotient will constitute all the prime factors of the given 13 number; 5 × 3 × 7 × 11 × 13 = 15015. Hence, the 11)143 = RULE. Divide the given number successively by its exact prime divisors, till a prime quotient is obtained; the divisors and the final quotient will be its prime factors. 61. There are no certain tests of the divisibility of numbers, except in a few cases, beyond which the divisors can be found only by trial. A number is exactly divisible By 2, if it is an even number. By 3, if the sum of its digits is exactly divisible by 3. By 4, if its two right-hand figures are ciphers, or express a number which is exactly divisible by 4. By 5, if it terminates with 0 or 5. By 6, if it is an even number of which 3 is an exact divisor. By 8, if its three right-hand figures are ciphers, or express a number which is exactly divisible by 8. By 9, if the sum of its digits is divisible by 9. |