131. There is an important relation between the highest
common factor and the lowest common multiple of two expressions which it is desirable to notice.
Let F be the highest common factor, and X the lowest common multiple of A and B. Then, as in the preceding article,
A = aF, B = bF, and X = abF.
Therefore the product AB = aF x bF = Fx abF = FX (1).
Hence the product of two expressions is equal to the product of their highest common factor and lowest common multiple.
Again, from (1) X = ^ = ^ x iJ = | x J ; hence the lowest common multiple of two expressions may be found by dividing their product by their highest common factor ; or by dividing either of them by their highest common factor, and multiplying the quotient by the other.
132. The lowest common multiple of three expressions A, B, C may be obtained as follows :
First find X, the L. C. M. of A and B. Next find T, the L. C. M. of X and C; then Y will be the required L. C. M. of A, B, a
For Y is the expression of lowest dimensions which is divisible by X and C, and X is the expression of lowest dimensions divisible by A and B. Therefore Y is the ex- pression of lowest dimensions divisible by all three.
EXAMPLES XII. c.
1. Find the highest common factor and the lowest common multiple of a:- - 5 .-« + 6, x^ - 4, x^ - 3 a; - 2. 2. Find the lowest common multiple of ab(x~ + 1) + x(«- -K b') and ab^x- - 1) + x{a- - b-). 3. Find the lowest common multiple of xy — hx, xy — ay, y'^ — Sby + 2 b'^, xy — 2bx — ay + 2 ab, xy — bx — ay + ab.
