Hence 11, being the last divisor, is the H. C. F. which is required. We may apply the same rule to algebraic expressions. The method is readily susceptible of proof. Suppose that we call the two quantities A and B, the degree of A being not lower than that of B. To determine their H. C. F. : Divide A by B. If there is a remainder divide B by that remainder, continue to make the divisor the dividend for each remainder left after dividing by that divisor, until there is no remainder. The last divisor is then the H. C. F. required. In this work the following observation should be kept in mind: 1. Each division should be continued until the remainder is of a lower degree than the divisor. 2. The work should be kept in the same order of powers of some common letter. 3. Either given expression may be divided by any quantity not a factor of the other, for such a quantity evidently cannot form a part of the H. C. F. Also, similarly, any remainder may be divided by any quantity not a common factor of the given expressions. 4. If the first term of any remainder be negative the sign of each term may be changed. 5. If the first term of the dividend or of any remainder is not divisible by the first term of the divisor, it may be made. divisible by multiplying the dividend or remainder by any quantity not a factor of the divisor. 6. If the quantities given have a common factor, which can be seen on inspection, remove it and ascertain the H. C. F. of the resulting expressions. The latter multiplied by the former is the H. C. F. of the quantities. -- Dividing 10x + 15 by 5 we have 2x 3. Remove from the former the common monomial factor of a and divide the latter by it, first multiplying the latter by 2, as 3 is not divisible by 2. As 2a is not divisible by 7a' we multiply each term of the expression by 7. To make the first term of this result of subtraction divisible by 7a' we multiply it by 7. To find the H. C. F. of three or more quantities, find the H. C. F. of two, then of this H. C. F. of two and another quantity, and so on. The last divisor will be the H. C. F. of all the quantities. LOWEST COMMON MULTIPLE. 45. A common multiple of two or more quantities is a quantity susceptible of being divided by every one of those quantities without a remainder. Thus it follows that a common multiple of several quantities must contain all the prime factors of every one of those several quantities. The lowest common multiple of two or more quantities is the product of their different prime factors, each of these prime factors being taken the greatest number of times which it occurs in any one of those quantities. Obviously, from this definition, the lowest common multiple of two or more quantities is the expression of lowest degree which can be divided by each of them without a remainder. For example, the lowest common multiple of x3y, x'y3, and y'z is x'y'z. When quantities are prime to each other, their product is their lowest common multiple. In the determination of the lowest common multiple of algebraic expressions, it is convenient to classify into the three following cases: Case I. When the quantities are monomials. Find the L. C. M. of 60a3y', 72a'x, and 84bx3. To the lowest common multiple of the coefficients annex all the letters which appear in the given quantities, giving each letter the highest exponent which it has in any of the given quantities. Case II. When the quantities are polynomials which can be readily factored by inspection. X3 9, and x3 6, x2 6 x = (x + 2) (x — 3). x2-9= (x + 3) (x-3). x3 — 6x2 + 9x = x (x+3)2. Hence the L. C. M. = x (x+3)2 (x + 2) (x-3). 3. m3 +n' and m2-n. Ans. (m2 — n3) (m2. mn +n3). Case III. When the quantities are polynomials which cannot be readily factored by inspection. Let A and B represent two expressions and let F be their highest common factor. Suppose that F is contained a times in A and b times in B. That is A = aF and B = bF, then As a and b can have no common factor, the L. C. M. of aF and bF is abF. Letting M represent the L. C. M. of A and B (1.e., aF and 6F) we have MabF. Multiply M by F we have Mx F, and multiplying abF by F we have abF2. or Therefore, FX MabF (2) Now A× B and F × M are both equal to abF'. Therefore, For, by (3), we have, that the product of any two quantities is equal to the product of their highest common factor and lowest common multiple. Therefore, to find the L. C. M. of two quantities: Divide their product by their highest common factor, or divide one of the quantities by the highest common factor of the two, and multiply the other quantity by the quotient thus obtained. Find the L. C. M. of 2x2 + x 6 and 4x2 8x + 3. By Article 44, their highest common factor is 2x-3 and 2x6 divided by 2x-3 is x 2, and 4x2 - 8x + 3 multiplied by x + 2 is 46. In arithmetic & signifies 2 divided by 3. Similarly the a expression signifies a divided by b. In other words, a units are divided into b parts and one of these equal parts taken, or one unit is divided into b equal parts and a parts taken. Such an expression as is termed a fraction. In such a frac a tion the part above the line (a) is called the numerator, as in |