VIII.

Common Factors and Multiples.

116. Common Factors.

A common factor of two or more integral numbers is an integral number which divides each of them without a remainder.

A common factor of two or more integral and rational expressions is an integral and rational expression which divides each of them without a remainder.

Thus $5a$ is a common factor of $20a$ and $25a$, $3x^2{y}^2$ is a common factor of $12x^2{y}^2$ and $15x^3{y}^3$.

Two numbers are said to be prime to each other when they have no common factor except $1$.

Two expressions are said to be prime to each other when they have no common factor except $1$.

The highest common factor of two or more integral numbers is the greatest number that will divide each of them without a remainder.

The highest common factor of two or more integral and rational expressions is an integral and rational expression of highest degree that will divide each of them without a remainder.

Thus $3a^2$ is the highest common factor of $3a^2$, $6a^3$, and $12a^4$, $5x^2{y}^2$ is the highest common factor of $10x^3{y}^2$ and $15x^2{y}^2$.

For brevity, we use H. C. F. for "highest common factor."

122. To Find the Highest Common Factor of Two or More Algebraic Expressions.

1. Find the H. C. F. of $42a^3{b}^2$ and $30a^2{b}^4$. {2} &42a^{3}b^{2} &&= 2 × 3 × 7 × aaa × bb; \ &30a^{2}b^{4} &&= 2 × 3 × 5 × aa × bbbb. \ \therefore\ &\text{the H.\,C.\,F.} &&= 2 × 3 × aa × bb, \quad\text{or}\quad 6a^{2}b^{2}.

2. Find the H. C. F. of $x^2-9y^2$ and $x^2+6xy+9y^2$. {2} &x^{2} - 9y^{2} &&= (x + 3y)(x - 3y); \ &x^{2} + 6xy + 9y^{2} &&= (x + 3y)(x + 3y). \ \therefore\ &\text{the H.\,C.\,F.} &&= (x + 3y).

3. Find the H. C. F. of $4x^2-4x-80$, $2x^2-18x+40$. {3} &4x^{2} -& 4x &- 80 &&= 4(x^{2} - x-20) \ &&& &&= 4(x - 5)(x + 4); \ &2x^{2} -& 18x &+ 40 &&= 2(x^{2} - 9x + 20) \ &&& &&= 2(x - 5)(x - 4). \ \therefore\ &\rlap{\text{the H.\,C.\,F.}} &&&&= 2(x - 5).

find the H. C. F. of two or more expressions, therefore,

Resolve each expression into its simplest factors.