637 143 65 18 the quotients, 4, 2, 5, being separated from the other figures. The reasoning, employed in the preceding example, may be applied to any numbers, and thus conduct us to this general rule. The greatest common divisor of two numbers will be found, by dividing the greater by the less; then the less by the remainder of the first division; then this remainder, by the remainder of the second division; then this second remainder by the third, or that of the third division ; and so on, till we arrive at an exact quotient; the last divisor will be the common divisor sought. 62. See two examples of the operation. 9024 3760 1504] 1752 75202 3008 21504 1504 752 00 752 then is the greatest common divisor of 9024 and 3760. By this last operation we see that the greatest common divisor of 957 and 47, is 1 only, that is, these two numbers, properly speaking, have no common divisor, since all whole numbers, like them, are divisible by 1. We may easily satisfy ourselves, that the rule of the preceding article must necessarily lead to this result, whenever the given numbers have no common divisor; for the remainders, each being less than the corresponding divisor, become less and less every operation, and it is plain, that the division will continue as long as there is a divisor greater than unity. 63. After these calculations, the fraction 143 and 3760 30249 can be at once reduced to their most simple term, by dividing the terms of the first by their common divisor, 13, and the terms of the second, by their common divisor, 752; we thus obtain and. As to the fraction,, it is altogether irreducible, since its terms have no common divisor but unity. 64. It is not always necessary to find the greatest common divisor of the given fraction; there are, as has before been remarked, reductions, which present themselves without this preparatory step. Every number terminated by one of the figures 0, 2, 4, 6, 8, is necessarily divisible by 2; for in dividing any number by 2, only 1 can remain from the ́tens; the last partial division can be performed on the numbers 0, 2, 4, 6, 8, if the tens leave no remainder, and on the numbers 10, 12, 14, 16, 18, if they do, and all these numbers are divisible by 2. The numbers divisible be 2 are called even numbers, because they can be divided into two equal parts. Also, every number terminated on the right by a cipher, or by 5, is divisible by 5, for when the division of the tens by 5 has been performed, the remainder, if there be one, must necessarily be either 1, 2, 3, or 4, the remaining part of the operation will be performed on the numbers 0, 5, 10, 15, 20, 25, 30, 85, 40, or 45, all of which are divisible by 5. The numbers, 10, 100, 1000, &c. expressed by unity followed by a number of ciphers, can be resolved into 9 added to 1, 99 added to 1, 999 added to 1, and so on; and the numbers 9, 99, 999, &c. being divisible by 3, and by 9, it follows that, if numbers of the form 10, 100, 1000, &c. be divided by 3 or 9, the remainder of the division will be 1. Now every number which, like 20, 300, or 5000, is expressed by a single significant figure, followed on the right by a number of ciphers, can be resolved into several numbers expressed by unity, followed on the right by a number of ciphers; 20 is equal to 10 added to 10; 300, to 100 added to 100 added to 100; 5000, to 1000 added to 1000 added to 1000 added to 1000 added to 1000; and so with others. Hence it follows, that if 20, or 10 added to 10, be divided by 3 or 9, the remainder will be 1 added to 1, or 2; if 300, or 100 added to 100 added to 100, be divided by 3 or 9, the remainder will be 1 added to 1 added to 1, or 3. In general, if we resolve in the same manner a number ex pressed by one significant figure, followed, on the right, by a number of ciphers, in order to divide it by 3 or 9; the remainder of this division will be equal to as many times 1, as there are units in the significant figure, that is, it will be equal to the significant figure itself. Now any number being resolved into units, tens, hundreds, &c. is formed by the union of several numbers expressed by a single significant figure; and, if each of these last be divided by 3 or 9, the remainder will be equal to one of the sig nificant figures of the given number; for instance, the division of hundreds will give, for a remainder, the figure occupying the place of hundreds; that of tens, the figure occupying the place of tens; and so of the others. If then, the sum of all these remainders be divisible by 3 or 9, the division of the given number by 3 or 9 can be performed exactly; whence it follows, that if the sum of the figures, constituting any number, be divisible by 3 or 9, the number itself is divisible by 3 or 9. Thus the numbers, 423, 4251, 15342, are divisible by S, because the sum of the significant figures is 9 in the first, 12 in the second, and 15 in the third. Also, 621, 8280, 934218, are divisible by 9, because the sum of the significant figures is 9 in the first, 18 in the second, and £7 in the third. It must be observed, that every number divisible by 9 is also divisible by 3, although every number divisible by 3 is not also divisible by 9. Observations might be made on several other numbers analogous to those just given on 2, 3, 5, and 9; but this would lead me too far from the subject. The numbers 1, 3, 5, 7, 11, 13, 17, &c. which can be divided only by themselves, and by unity, are called prime numbers; two numbers, as 12 and 35, having, each of them, divisors, but neither of them any one, that is common to it with the other, are called prime to each other. Consequently, the numerator and denominator of an irreducible fraction are prime to each other. Examples for practice under Article 61. What is the greatest common divisor of 24 and 36? Ans. 12. What is the greatest common divisor of 35 and 100? Ans. 5. What is the greatest common divisor of 312 and 504 ? Ans. 24. Examples for practice under articles 57, 58, and 60. 65. After this digression we will resume the examination of the table in article 55, multiplied, By multiplying the numerator, the fraction is {divided, By dividing By dividing } } that we may deduce from it some new inferences. divided, multiplied, We see at once, by an inspection of this table, that a fraction can be multiplied in two ways, namely, by multiplying its numerator, or dividing its denominator, and that it can also be divided in two ways, namely, by dividing its numerator, or multiplying its denominator; hence it follows, that multiplication alone, according as it is performed on the numerator or denominator, is sufficient for the multiplication and division of fractions by whole numbers. Thus, multiplied by 7 units, makes 2}; 4, divided by 3, makes 7. 66. The doctrine of fractions enables us definition of multiplication given in article 21. to generalize the When the multi "plier is a whole number, it shows how many times the multiplicand is to be repeated; but the term multiplication, extended to fractional expressions, does not always imply augmentation, as in the case of whole numbers. To comprehend in one statement every possible case, it may be said, that to multiply one number by another is, to form a number by means of the first, in the same manner as the second is formed, by means of unity. In reality, when it is required to multiply by 2, by S, &c. the product consists of twice, three times, &c. the multiplicand, in the same way as the multiplier consists of two, three, &c. units; and to multiply any number by a fraction, for example, is to take the fifth part of it, because the multiplier, being the fifth part of unity, shows that the product ought to be the fifth part of the multiplicand*. Also, to multiply any number by is to take out of this number or the multiplicand, a part, which shall be four fifths of it, or equal to four times one fifth. Hence it follows, that the object in multiplying by a fraction, whatever may be the multiplicand, is, to take out of the multiplicand a part, denoted by the multiplying fraction; and that this operation is composed of two others, namely, a division and a multiplication, in which the divisor and multiplier are whole numbers. Thus, for instance, to take of any number, it is first necessary to find the fifth part, by dividing the number by 5, and to repeat this fifth part four times, by multiplying it by 4. We see, in general, that the multiplicand must be divided by the denominator of the multiplying fraction, and the quotient be multiplied by its numerator. The multiplier being less than unity, the product will be smaller than the multiplicand, to which it would be only equal, if the multiplier were 1. 67. If the multiplicand be a whole number divisible by 5, for * We are led to this statement, by a question which often presents itself; namely, where the price of any quantity of a thing is required, the price of the unity of the thing being known. The question evidently remains the same, whether the given quantity be greater or less than this unity. |