Page:Elementary algebra (1896).djvu/120

102 Rh CHAPTER XII.

Lowest Common Multiple.

125. Definition. The Lowest Common Multiple of two or more algebraic expressions is the expression of lowest dimensions which is divisible by each of them without remainder.

The abbreviation L. C. M. is sometimes used instead of the words lowest common multiple.

SIMPLE EXPRESSIONS.

126. The L. C. M. can be written by inspection.

Ex. 1. The lowest common multiple of a^4, a^3, a2, a6 is a^6.

Ex. 2. The lowest common multiple of a^3b^4, ab5, a^2b7 is a^3b^7 ; for a3 is the lowest power of a that is divisible by each of the quantities a^3, a, a2. and b^7 is the lowest power of b that is divisible by each of the quantities b4, b^5, b7.

127. If the expressions have numerical coefficients, find by Arithmetic their least common multiple, and prefix it as a coefficient to the algebraic lowest common multiple.

Ex. The lowest common multiple of 21 a^4x3y, 35 a2x^4y, 28 a3xy^4 is 420 a^4x^4y^4 ; for it consists of the product of

(1) the numerical least common multiple of the coefficients ;

(2) the lowest power of each letter which is divisible by every power of that letter occurring in the given expressions.

EXAMPLES XII. a.

Find the lowest common multiple of

1. x^3y2, xyz. 4. 12ab,8xy. 7. 2x,3y,4z. 2. 3x2yz,4x^3y3 5. ac, bc, ab. 8. 3x^2,4 y2, 3z^2 3. 5 a2bc3, 4 ab2c. 6. a^2c, bc2, cb^2. 9. 7 a2, 2 ab, 3 b2.