Page images
PDF
EPUB

Greatest Common Divisor.

by dividing them by a common factor or divisor. But when they have no common divisor, the fraction is said to be in its lowest terms.

A fraction is, consequently, reduced to its lowest terms, by dividing its terms by their greatest common factor or divisor.

57. Problem. To find the greatest common divisor of several monomials.

Solution. It is equal to the product of the greatest common divisor of the coefficients, by those different literal factors which are common to all the monomials, each literal factor being raised to the lowest power which it has in either of the monomials.

58. EXAMPLES.

1. Find the greatest common divisor of 75 a3 b8 c d11 x9 and 50 a3 c2 d11 x5.

Ans. 25 a3 c d11 x5.

[blocks in formation]

59. Lemma. The greatest common divisor of two quantities is the same with the greatest common divisor of the least of them, and of their remainder after division.

Demonstration. Let the greatest of the two quantities be A, and the least B; let the entire part of their quotient after division be Q, and the remainder R; and let the greatest

Greatest Common Divisor.

common divisor of A and B be D, and that of B and R be E. We are to prove that

D= E.

Now since R is the remainder of the division of A by B, we have

R-A-B.Q;

and, consequently, D, which is a divisor of A and B, must divide R; that is, D is a common divisor of B and R, and cannot therefore be greater than their greatest common divisor E.

[blocks in formation]

and, consequently, E, which is a divisor of B and R, must. divide A; that is, E is a common divisor of A and B, and cannot therefore be greater than their greatest common divisor D.

D and E, then, are two quantities such that neither is greater than the other; and must therefore be equal.

60. Problem. To find the greatest common divisor of any two quantities.

Solution. Divide the greater quantity by the less, and the remainder, which is less than either of the given quantities, is, by the preceding article, divisible by the greatest common divisor.

In the same way, from this remainder and the divisor a still smaller remainder can be found, which is divisible by the greatest common divisor; and, by continuing this process with each remainder and its corresponding divisor, quantities smaller and smaller are found, which are all divisible by the greatest common divisor, until at length the common divisor itself must be attained.

Greatest Common Divisor.

The greatest common divisor, when obtained, is at once recognised from the fact, that the preceding divisor is exactly divisible by it without any remainder.

The quantity thus obtained, must be the greatest common divisor required; for, from the preceding article, the greatest common divisor of each remainder and its divisor is the same with that of the divisor and its dividend, that is, of the preceding remainder and its divisor; hence, it is the same with that of any divisor and its dividend, or with that of the given quantities.

61. Corollary. When the remainders decrease to unity, the given quantities have no common divisor, and are said to be incommensurable or prime to each other.

62. EXAMPLES.

1. Find the greatest common divisor of 1825 and 1995.

[blocks in formation]

Greatest Common Divisor.

This process may be written more neatly and concisely as follows.

[blocks in formation]

2. Find the greatest common divisor of 13212 and 1851.

Ans. 3.

3. Find the greatest common divisor of 1221 and 333.

Ans. 111.

63. The above rule requires some modification in its application to polynomials.

Thus it frequently happens in the successive divisions, that the term of the dividend, from which the term of the quotient is to be obtained, is not divisible by the corresponding term of the divisor. This, sometimes, arises from a monomial factor of the divisor which is prime to the dividend, and which may be suppressed.

For, since the greatest common divisor of two quantities is only the product of their common factors, it is not affected by any factor of the one quantity which is prime to the other.

Hence any monomial factor of either dividend or its divisor is to be suppressed which is prime to the other of these two quantities, and when there is such a factor it is readily obtained by inspection.

Greatest Common Divisor.

But if, after this reduction, the first term of the dividend, when arranged according to the powers of some letter, is still not divisible by the first term of the divisor similarly arranged; it follows from the preceding reasoning that it can lead to no error to

Multiply the dividend by some monomial factor which will render its first term divisible by the first term of the divisor, and which is prime to the reduced divisor. Such a factor can always be obtained by simple inspection.

When the given quantities have any common monomial factor it is easily obtained from inspection, and it should be suppressed at first, and afterwards multiplied by the greatest common divisor of the remaining polynomials.

Since any quantity which is divisible by A is also divisible by ·A; and any quantity which is divisible by - A is also divisible by A;

[ocr errors]

All the signs of any divisor may be reversed at pleasure.

64. EXAMPLES.

1. Find the greatest common divisor of 6 a2 x3+21 a3 x2 27 a5 and 4 x4 +5 a2 x2 + 21 a3 x.

Solution. These quantities have no common monomial factor; but the monomial factor 3 a2 common to all the terms of the first of them, and the factor x common to all the terms of the second, being suppressed in columns 1 and

« PreviousContinue »