Lesson 2 · from §59 of the 1910 text
Lowest Common Multiple
The lowest common multiple (L.C.M.) of two or more expressions is the expression of lowest degree that each of them divides exactly.
Example. Find the L.C.M. of 24x^3y^2, 36x^4y, and 54x^2y^2z. Factor: 2^3\cdot3\,x^3y^2, \;2^2\cdot3^2x^4y, \;2\cdot3^3x^2y^2z. The L.C.M. needs the highest power of every factor: 2^3\cdot3^3x^4y^2z = 216x^4y^2z.
Example. 9a^2y + 18aby + 9b^2y = 3^2y(a + b)^2, \;15ax^2 - 15bx^2 = 3\cdot5x^2(a - b), and 2a^2 - 4ab + 2b^2 = 2(a - b)^2. The L.C.M. is 2\cdot3^2\cdot5\,x^2y(a + b)^2(a - b)^2.
Rule. Separate each expression into its prime factors. Then find the product of all the different prime factors, using each the greatest number of times it occurs in any one expression.
You'll use the L.C.M. constantly in the next chapter: it is the lowest common denominator for adding fractions.
Quick check
24x^3y^2, 36x^4y and 54x^2y^2z
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Numbers first.
12 and 18
32 and 48
20 and 28
96 and 144
64, 120 and 216
128, 160 and 200
Monomials.
x^2y, xy^2 and xy^3
6cd^3, 4c^5de and 10c^2d^2e^3
8abc, 3b^2c and 12c^2
18m^2, 15mn^2p and 20m^3p^2
20, 18xy^4 and 27ax^3y
36ab^2, 42abc and 63b^3c
Polynomials. Factor everything first. A factored answer like x(x - 3)(x + 3) is the easy way to type it.
4a and a^2-ab
12ax and 3a^3x^2-3ax^3
cx+cy and dy+dx
3x+3z and 6a^2x+6a^2z
x^2-xy and ax+ay
x^2-9 and x^2-5x+6
c^2-4 and c^2-8c-20
4ax, 4x^2-1 and 4x^2+4x+1
x^2+1, x^4-1 and x^4-2x^2+1
4-c^2, c^3+8 and c^2+6c+8
ac-2bd+2ad-bc and a^2+ab-2b^2
x^2-y^2, x^3-y^3 and x^2+2xy+y^2
ax^2+bxy, 2ax+2by and a^2x^2-b^2y^2
2x^3-2x, 3x^4+15x^3-18x^2 and x^2-36
8-y^3, y^2-4 and 4y^2+2y^3+y^4
2x^3-6x^2+4x^2y and x^2-4y^2-6x+9