triple, &c. is taken; in short it is multiplied by the same number, by which it was at first divided. Thus the fraction is equal to 1, and is equal to .

58. It is not with fractions as with whole numbers, in which a magnitude, so long as it is considered with relation to the same unit, is susceptible of but one expression. In fractions on the contrary, the same magnitude can be expressed in an infinite number of ways. For instance, the fractions

in each of which the denominator is twice as great as the numerator, express, under different forms the half of an unit. The fractions

of which the denominator is three times as great as the numerator, represent each the third part of an unit. Among all the forms, which the given fraction assumes, in each instance, the first is the most remarkable, as being the most simple; and, consequently, it is well to know how to find it from any of the others. It is obtained by dividing the two terms of the others by the same number, which, as has already been shown, does not alter their value. Thus if we divide by 7 the two terms of the fraction, we come back to; and, performing the same operation on 7 we get 1.

59. It is by following this process, that a fraction is reduced to its most simple terms; it cannot, however, be applied, except to fractions, of which the numerator and denominator are divisible by the same number; in all other cases the given fraction is the most simple of all those, that can represent the quantity it expresses. Thus the fractions, 2, 18, the terms of which cannot be divided by the same number, or have no common divisor, are irreducible, and, consequently, cannot express, in a more simple manner, the magnitudes which they represent.

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60. Hence it follows, that to simplify a fraction, we must endeavour to divide its two terms by some one of the numbers, 2, 3, &c.; but by this uncertain mode of proceeding it will not

be always possible to come at the most simple terms of the given fraction, or at least, it will often be necessary to perform a great number of operations.

If, for instance, the fraction

were given, it may be seen at once, that each of its terms is a multiple of 2, and dividing them by this number, we obtain 12; dividing these last also by 2, we obtain. Although much more simple now than at first, this fraction is still susceptible of reduction, for its two terms can be divided by 3, and it then becomes .

If we observe, that to divide a number by 2, then the quotient by 2, and then the second quotient by 3, is the same thing as to divide the original number by the product of the numbers, 2, 2, and 3, which amounts to 12, we shall see that the three above operations can be performed at once by dividing the two terms of the given fraction by 12, and we shall again have

The numbers 2, 3, 4, and 12, each dividing the two numbers 24 and 84 at the same time, are the common divisors of these numbers; but 12 is the most worthy of attention, because it is the greatest, and it is by employing the greatest common divisor of the two terms of the given fraction, that it is reduced at once to its most simple terms. We have then this important problem to solve, two numbers being given, to find their greatest common divisort.

61. We arrive at the knowledge of the common divisor of two numbers by a sort of trial easily made, and which has this recommendation, that each step brings us nearer and nearer to the number sought. To explain it clearly, I will take an example.

Let the two numbers be 637 and 143. It is plain, that the greatest common divisor of these two numbers cannot exceed the smallest of them; it is proper then to try if the number 143, which divides itself and gives 1 for the quotient, will also divide the number 637, in which case it will be the greatest common divisor sought. In the given example this is not the case; we obtain a quotient 4, and a remainder 65.

What is here called the greatest common divisor, is sometimes called the greatest common measure.

Now it is plain, that every common divisor of the two numbers, 143 and 637, ought also to divide 65, the remainder resulting from their division; for the greater, 637, is equal to the less, 143, multiplied by 4, plus the remainder, 65, (50); now in dividing 637 by the common divisor sought, we shall have an exact quotient; it follows then, that we must obtain a like quotient, by dividing the assemblage of parts, of which 637 is composed, by the same divisor; but the product of 143 by 4 must necessarily be divisible by the common divisor, which is a factor of 143, and consequently the other part, 65, must also be divisible by the same divisor; otherwise the quotient would be a whole number accompanied by a fraction, and consequently could not be equal to the whole number, resulting from the division of 637 by the common divisor. By the same reasoning, it may be proved in general, that every common divisor of two numbers must also divide the remainder resulting from the division of the greater of the two by the less.

According to this principle, we see, that the common divisor of the numbers 637 and 143, must also be the common divisor of the numbers 143 and 65; but as the last cannot be divided by a number greater than itself, it is necessary to try 65 first. Dividing 143 by 65, we find a quotient 2, and a remainder 13; 65 then is not the divisor sought. By a course of reasoning, similar to that pursued with regard to the numbers, 637, 143, and the remainder, resulting from their division, 65, it will be seen that every common divisor of 143 and 65 must also divide the numbers 65 and 13; now the greatest common divisor of these two last cannot exceed 13; we must therefore try, if 13 will divide 65, which is the case, and the quotient is 5; then 13 is the greatest common divisor sought.

We can make ourselves certain of its possessing this property by resuming the operations in an inverse order, as follows;

As 13 divides 65 and 13, it will divide 143, which consists of twice 65 added to 13; as it divides 65 and 143, it will divide 637, which consists of 4 times 143 added to 65; 13 then is the common divisor of the two given numbers. It is also evident, by the very mode of finding it, that there can be no common divisor greater than 13, since 13 must be divided by it.

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It is convenient in practice, to place the successive divisions

one after the other, and to dispose of the operation, as may be seen in the following example;

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

7520 2 3008 21504 2

752

752 then is the greatest common divisor of 9024 and 3760.

9371 47 44 3 21 1

47 19 44 13 14 2 1 2 2

By this last operation we see that the greatest common divisor of 937 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 14 and $79, can

90241

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be at once reduced to their most simple terms, 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, 7, 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 by 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, 35, 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 Arith.

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