108. In dividing one number by another, we seek a number which, when multiplied by the divisor, reproduces the dividend. Thus, as shown in the last example, may Multiplication be made as subservient in proving Division, as Division has hitherto been in proving Multiplication, 109. All the sums which we have had hitherto to divide have been the products of two factors, and being divided by one of these factors, the quotient was necessarily the other factor. But it frequently happens, that in dividing one number by another, the divisor is not contained in the dividend an exact number of times, but that, after we have pursued the division as far as we can, there remains some part of the divi- . dend which is less than the divisor, and therefore cannot be divided by it. In cases of this kind, the remainder is added to the quotient, with a line drawn under it, and the divisor placed underneath in the form of a fraction. In proving the question by Multiplication, the remainder is added in with the product, as shown in the following examples: : 110. In cases like the annexed, where the bringing down of one figure to the remainder is insufficient to make an amount equal to the divisor, we write a cypher in the quotient and bring down another figure, and as the divisor is now contained twice, we write 2 in the quotient as before directed. 834 834)85079( 102 834 83411 1679 1668 85079 834 111. When the quotient is less than the divisor, as in the annexed example, it is frequently best, in proving it by Multiplication, to multiply by the quotient rather than by the divisor; and as in proving Division questions by Multiplication, the workings are apt to run into one another, and become confused, it is sometimes convenient to begin the multiplication by the left hand figure, as in this example. To multiply by 100, the student knows, is but to annex two cyphers to the right hand of the number to be multiplied. In this example, the remainder 11 is added in, and properly occupies the place of the two cyphers; the divisor is then multiplied by 2, as shown in the example. 112. In the annexed example, though the quotient is larger in amount than the divisor, it is yet better to multiply by the quotient, as by so doing, we can perform the operation in two lines; whereas, if we multiply the quotient by the divisor, it will require three lines. 113. In the following example, the 652 652)587532(901 586880 732 652 652 587532 80 114. In the above example also, though the quotient contains four figures and the divisor but three, yet there is a great saving both in time and figures in multiplying by the quotient. 1006 700+ 130 7043301006 × 700 + 130 20121006 X 2 7023181006 698 115. In cases where the multiplier is so many hundreds, and near another hundred, the operation may be shortened, as in the annexed example, by adding as much to the multiplier as will raise it to a certain number of hundreds : then multiplying by the multiplier thus increased, and from the product deducting the product of the multiplicand by the amount which it requires to raise the multiplier to the even hundreds. Thus in the above example, instead of multiplying by 698, we multiply by 700, and from the product deduct twice the multiplicand; the difference, it will be seen, is equal to the product of 1006 by 698, or (1006 × 698) + 130=(1006 × 700) + 130 — (1006 × 2). 9. What number is it, which, when divided by 396, the quotient will be 620 ? 10. What number is it in which 7672 is contained 597 times ? 11. What number is it which, when multiplied by 6241, the product will be 528188312? 12. The product of two factors being 3325060, and one of the factors being 4805, what is the other factor? 13. By what number can we divide 3339475, so as to give us 695 for a quotient ? 13. What number is it by which, if we multiply 4805, we shall obtain 3339475 for a product? 14. How many pence are equal to 3672 pounds, one pound being equal to 240 pence ? 15. How many farthings are equal to 3695 pounds, one pound being equal to 960 farthings? 16. How many shillings are equal to 8650 pounds, 20 shillings being equal to one pound? 17. In 367800 sixpences how many pounds, there being 40 sixpences in one pound? 18. In 3695848 pounds weight how many hundred-weights, there being 112 pounds in one hundred-weight? 19. In 360 barrels of porter, each barrel containing 36 gallons, how many pints, when each gallon contains 8 pints? 20. The distance between Newcastle and Edinburgh is 117 miles, how many strides would a man take between the two places, supposing him to cover a yard at a stride, a mile being 1760 yards? CASE 5. 117. When the student has become tolerably ready at dividing, (which he must be by this time, if he has paid proper attention to what has been advanced for his instruction), I recommend that he performs the Subtraction as he proceeds with the Multiplication, and that he notes down the remainder only. By this method many figures are dispensed with, and the operation much shortened. The student, however, should not attempt dividing by this, unless he be pretty expert at the other method, else he may be liable to commit errors; for though it is at all times desirable to shorten operations as much as possible, yet accuracy is the chief thing to be attended to. 134)6743685-( 50326 436 118. Observe the working of the following example. After selecting the first portion of figures of the dividend in which the divisor is contained, we place 5 in the quotient, and say 5 times 4 are twenty, and twenty from twenty-four, and 4. We set down 4, and go two; 5 times 3 are fifteen, and two carried are seventeen; seventeen from seventeen, and nothing, we set a dot, and go one; 5 times 1 are five, and one carried is six; six from six, and nothing; we then bring down the 3, but 134 is not contained in 43, therefore we set 0 in quotient; we then bring down the 6. The divisor is now con 134 201305 ⚫348 150978 50326 • 805 6743685 tained three times; we therefore set 3 in the quotient, and say 3 times 4 are twelve; twelve from sixteen, and 4; we set own 4, and go one; 3 times 3 are nine, and I carried are ten; ten from thirteen, and three; we set down 3, and go onè; 3 times one is three, and one carried are four; four from four, and nothing. It will be unnecessary to proceed further with the explanation; from what has been said the student will readily see the mode of operation. 463)5396472( 11655 119. One other example I shall exhibit ready performed, and then leave the student to exercise himself by performing the rest of the examples in paragraph 116. CASE 6. Of Prime Numbers, of Multiples, of Sub-multiples, and of the Method of Performing Short Division. 120. At paragraph 89, it is stated, that when I came to treat of Prime Numbers and Multiples, I should then show a method of performing Division, which, in its process, is much shorter and easier than the method we had then under consideration; but that this method, which is called Short Division, is only applicable to questions, the divisors of which are divisible, that is, multiples of other numbers. When the divisor is a prime number, the only way of peforming the operation is by Long Division. 121. In order therefore to enable the student to determine whether a question can be performed by Short Division or not, it is necessary he should have some knowledge of prime numbers and multiples. 122. A PRIME NUMBER is a number which cannot be divided by any other number except 1, without leaving a remainder. 123. Every number is either prime itself, or it is reducible inte a certain number of primes. 5 and 7 are prime numbers, because they are not divisible by any number except 1, but their product 35 is a divisible number, being divisible into 7 parts of 5 each, or into 5 parts of 7 each, which parts are prime numbers; and therefore, if a number be the product of two prime numbers, such number is divisible only by its two factors. 124. We may determine whether a number be prime or not by dividing it by all the numbers less than its half; it being evident that no number can * Р ROOTS 21 22 23 24 25 26 27 28 29 30 31 32 SQUARES 4 9 ROOTS 72 P3 TABLE OF SQUARES AND ROOTS. 16 25 36 49 64 81 100 121 144 169 196 225 256, 289 324 361 Р Thus we find that 23 cannot be divided by any number less than its half, which is between 11 and 12; it is, therefore, a prime number. If numbers be large, however, this process of discovering whether they are prime or not, would be attended with much labour and loss of time. But there is another mode, by which this may be ascertained, less liable to these objections, which is this:-Divide the given number by all the prime numbers, that are not greater than its SQUARE ROOT, and if it cannot be divided by any of these without a remainder, it is a prime number. Thus in the number 103 we perceive it is a prime number because it is not evenly divisible by 7, 5, 3, or 2, which are the only prime numbers not greater than its Square Root which lies between 10 and 11. To understand the reason of this clearly, the student should know something of Algebra; the reason, therefore, cannot be given in the present stage of his acquirements. He may easily, however, satisfy himself that what is her Р 33 34 35 37 38 39 Р Р Р 5 6 7 8 9 10 11 12 13 14 15 16 17 advanced is a fact, by trying a number of examples, and if The Square Root of a number is a number which when multiplied into itself produces the square, thus-3 is the square root of 9, and 9 is the square of 3, because 3 × 3 = 9. |