137. Common Factors. A common factor of two or more
integral numbers in Arithmetic is an integral number that
divides each of them without a remainder.
188. A common factor of two or more integral and rational
expressions in Algebra is an integral and rational expression
that divides each of them without a remainder.
Thus, 6 a is a common factor of 20 a and 26 a.
189. Two numbers in Arithmetic are said to be prime to
each other when they have no common factor except 1.
140. Two expressions in Algebra are said to be prime to
each other when they have no common factor except 1.
141. The greatest common factor of two or more integral
numbers in Arithmetic is the greatest number that will
divide each of them without a remainder.
142. The highest common factor of two or more integral
and rational expressions in Algebra is an integral and
rational expression of highest degree that will divide each
of them without a remainder.
Thus, 3 a2 is the highest common factor of 3 a^, 6 a^, and 12 a* ;
6x*y* is the highest common factor of lOx^yS and Ibx^^,



Digitized by CjOOQ IC



COMMON FACTORS AND MULTIPLES. 109

For brevity, we use H. C. F. for highest common factor.
1. Find the H. C. F. of 42 a^b^ and 30 a%*.
42a^P =2x3x7XaaaXWj

SOa^b^ =2x3x5XaaX bbbb.
.-. the H.C.F. = 2x3XaaX^^ = 6a*b\

2. Find the H.C.F. of x^ - 9y^ and x» + 6xy + 9y*.
x^-dy^ =(x-hSy)(x-3y)i

x^ + 6xy + 9y« =^(x + Sy)(x + 3y).
/.theH.C.F.=a; + 3y.
3. FindtheH.C.F. of

4««-4x-80; 2x^-lSx + A0; 2 «« - 24 « + 70.
