528 ARITHMETIC multiplicand and multiplier are also said to be factors of the product. The multiplicand might be written down the required number of times, and the sum found by addition. But this tedious process is unnecessary when the numbers to be added are all the same We do not then require to pass from figure to figure, as when they are different. In adding six nines, for instance, we do not need to proceed step by step (as with 6 + 5 + 9 + 8 + 6 + 7 = 41), but know the result, 54, at once from a table of products which is committed to memory. When two numbers are to be multiplied together, either may be taken as the multiplier. For it is evident that the four rows of dots, five in a row, in the margin, are the same as the five rows, four in a row accord ing as we take the rows horizontally or vertically; that is, 4 times 5 and 5 times 4 amount to the same number. The process of multiplication by a single digit by 8, for instance is nothing but an abridgement of the operation of writing the multiplicand eight times and adding. When there are more digits than one in the multiplier, we arrange the successive products in the well-known fashion, because the multiplication has to be by 10 times the second digit (counting from the right), 100 times the third, and so on. Thus, in multiplying 92058 by 734, the 2 of the multipli cand is really 2000 ; the 3 of the multiplier, 30 ; and the product of these, 6, i.e., 60,000, falls by the ordinary pro cess into its proper place. The process of working may sometimes be considerably shortened by multiplying a product already obtained. Thus, to multiply by 568, we may multiply first by 8, and then take 7 times the result, obtaining the product by 56 at once. So with 549378, we may begin with 9, and mul tiply the 9 product by 6 for 54, and the 54 product by 7 for 378, observing to arrange the lines correctly, by placing under the right-hand figure of every multiplier the right- hand figure of the corresponding product. 7. Division is the method of finding how often one given number contains another. Of these two numbers, the former is called the dividend, and the latter the divisor. The number expressing the times that the first contains the second is called the quo tient. When the number of times is not exact, the excess of the dividend over the divisor taken the greatest number of times that the dividend contains it exactly is called the remainder. As multiplication is a short method of addition, division (which is the converse of multiplication) is an abridged subtraction. Were we to subtract the divisor from the dividend, subtract it again from the remainder, and con tinue the process till a remainder less than the divisor were obtained, the number of subtractions would give the re quired quotient. But this operation is greatly shortened by means of multiplication. Thus, if it is required to find how often 9 is contained in 49, remembering that 5x9 is 45, and that this is the nearest product not greater than 49, we have at once the quotient 5 and remainder 4. In dividing, e.g., 167685 by 287, by the ordinary pro cess of long division, we find (after, it may be, a trial or two) that 1676 is more than 5 times but less than 6 times 287. The first quotient figure is therefore 5, represent ing 500 and proceeding in the customary way, we take from the dividend 500 times the divisor, then 80 times, and then 4 times, with 77 over. The dividend, therefore, contains the divisor 584 times, with a remainder of 77. When the divisor does not exceed 1 2, the operation is ciuducted mentally, and the quotient set down at once. This is called short division. If the divisor is made up of factors not greater than 12, short division may be employed, the factors being taken in succes- sion. When there are two such divisors, the remainder of the whole division is obtained 8)725 2 from the partial remainders, by multiplying g(V 5 the second of these by the first divisor, and adding the first to the product. Thus, in the annexed example, division by 48 gives a remainder of 32, for it will be seen that 725 is 5 more than 8 times 90, and 4352 is 2 more than 6 times 725. Therefore, 4352 is 2 more than the sum of 48 times 90 and 6 times 5, i.e., it exceeds 48 times 90 by 6 x 5 + 2 = 32, or contains 48 90 times and 32 over. Measures and Multiples of Numbers. 8. A measure of any number is a number that divides it without a remainder. A imdtiple of any number is a number that it divides with out a remainder. A common measure, or common multiple of several numbers, is a number which is a measure or a multiple of each of them. Thus, 3 is a common measure of 12, 18, and 24; 60 is a common multiple of 6. 10, and 15. A measure of a number is sometimes called a sub- multiple of it. A. prime number is a number which no other, except unity, divides without a remainder; as 2, 3, 5, 7, 11, 13, 17, &c. Numbers which are divisible by other numbers without remainder, that is, which can be resolved into factors, are called composite numbers; as 4, 6, 8, 9, 10, &c. Any factors into which a composite number can be divided are called its component parts. Numbers are said to be prime to each other when they have no common measure, as 15 and 28. The prime factors of a number are the prime numbers of which it is the continued product. Thus, 2, 3, 7 are the prime factors of 42 ; 2, 2, 3, 5, of 60. 9. To find the greatest common measure of two given numbers, the greater number is divided by the less ; the former divisor is then divided by the remainder, and each successive divisor by the remainder obtained in dividing by it, till there is no remainder. The last divisor is the greatest common measure. This depends on the two following principles : (1.) If a number measures any other, it measures every multiple of that other; for obviously, since 7 measures 56, it also measures 12 times or 17 times 56; and, (2.) Every number that is a common measure of two others measures also their sum or their difference ; for the sum or difference of, say, 13 times 8 and 22 times 8, must, it is evident, be some multiple of 8. Thus, to find the greatest common measure of 475 ar.d 589, dividing 589 by 475, we have the remainder 114 ; dividing 475 by 114, we have the remainder 19 ; and 114 divided by 19 leaves no remainder. Therefore, 19 is the greatest common measure. For any number that measures 589 and 475 will measure their difference, 114, and will, therefore, measure 456, which is a multiple of 114. Also, any common measure of 475 and 456 will measure their difference, 19. Therefore, no number greater than 19 can measure both 589 and 475. Again, 19 will measure both, for it measures 114, and therefore measures 456, a multiple of 114. Hence it measures 475, which is 456 + 19, and also 589, which is 475 + 114. Therefore, 19 measures both mimbers, and since no greater number does so, it is- their greatest common measure. In seeking for measures or factors, it will be of advantage to attend to the following properties of numbers. (For the sake of brevity, we use " divisible " here for " divisible without remainder.") (1.) A number is divisible by 2, if the last digit is divisible by 2 ; by 4, if the last two digits are divisible by 4 ; and by 8, if the last three digits are divisible by 8. For, to take the last case, the figures preceding the last Measure and multiple Creates coinmor
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