Unit-
8
Highest
Common Factor
and
Lowest
Common Multiple
Exercise with Answers
The highest common
factor of the given two or more algebraic expressions is an expression of the
highest degree which is common to all given expressions that divides each given
expression without any remainder. It is written in the short form as H.C.F.
If there is no common
factor among the given expressions then the H.C.F of the given expressions is
1.
1.
Find the H.C.F. of:
a.
4x2y and
12xy3 (Ans 4xy)
b.
12x3y2
and 18x2y5 (Ans 6x2y2)
c.
a2bc3
and a3b2c - a2bc (Ans 2x + 1)
d.
15x2y3,
40x3y4
and 55x3y5
- 5x2y3 (Ans x + y)
2.
Find the H.C.F. of:
a.
a3b – ab3 and a2 + 2ab
+ b2 (Ans
a + b)
b.
x2 -9
and x2 – x - 6 (Ans x - 3)
c.
4x3 – x and 4x2 + 4x +
1 (Ans 2x + 1)
d.
x2y – y3
and x3 + y3 (Ans x + y)
3.
Find the H.C.F. of:
a.
x3 – y3
and x4 + x2y2
+y4 (Ans x2 + xy + y2)
b.
a3 + 1 and a4 + a2
+1 (Ans a2
- a + 1)
c.
8m3 +n3 and 16m4
+ 4m2n2 +n4
(Ans 4m2
- 2mn + n2)
d.
x4 + 1 +
1/
x4 and x3 – 1/ x3 (Ans x2 + 1 +
1/ x2)
4.
Find the H.C.F. of:
a.
2a2 – 8, a2 – a - 2 and a4
-8a (Ans a - 2)
b.
3x2 – 8x + 4, x4 - 8x and x2 –
4 (Ans x - 2)
c.
2m3 + 16, m2 +4m + 4 and
m2 +3m + 2 (Ans
m + 2)
d.
4y3 – y, 2y3 – y2
– y and 8y4 +y (Ans {y(2y + 1)}
5.
Find the H.C.F. of:
a.
x3 – y3,
x6 – y6
and x4 + x2y2
+ y4 (Ans x2 + xy + y2)
b.
x3 – 1, x4 + x2 + 1
and x6 - 1 (Ans x2 + x + 1)
c.
a3
+ b3, a6 – b6 and a4 + a2b2
+ b4 (Ans a2 - ab + b2)
d.
x3 + 1, x4 + x2 + 1
and x3 - 1 -
2x2 + 2x (Ans x2 - x + 1)
6.
Find the H.C.F. of:
a.
a2 + 2ab + b2 – c2,
b2 + 2bc + c2 – a2 and c2 +2ac + a2
– b2 (Ans a + b + c)
b.
x6 – 1, x4 + x3 + x2 and x3 + 2x2 + 2x +
1 (Ans x2 + x + 1)
c.
2x2 – 3x – 2,
8x3 + 1
and 4x2 – 1 (Ans 2x + 1)
d.
2y3 -16, y2 + 3y + 2 and
2y2 - 8 (Ans1)
e.
4x 3 - 6x 2y + 9x y2,
16x4 + 36x2y2
+ 81y4 and 8x 3 + 27y3
(Ans 4x2 - 6xy + 9y2)
The lowest common multiple of
the given two algebraic expressions is the product of the H.C.F. and the
remaining factors. Lowest common multiple of two or more than two algebraic expressions is an
expression of the least degree which is exactly divisible by the given
algebraic expressions. It is written as L.C.M in the short from.
1.
Find the L.C.M. of:
a.
2x2 and
3y (Ans 6x2y)
b.
6x2y and
15xy2z
(Ans 30x2y2z)
c.
a2b, ab3 and 2a2b4x (Ans 2a2b4x)
d.
15a2b3, 40a4b5
and 60a3b2c (Ans120a4b5c)
2.
Find the L.C.M. of:
a.
2x2 – 8y2
and x2 – xy – 2y2 {Ans (x+y) (x2-4y2)}
b.
(b) 3m2 – 27 and m2 + m-
6 {Ans
3(m-2)(m2-9)}
c.
x3y – xy3
and x2 + 2xy +y2
{Ans xy(x+y)2
(x -y)}
d.
a2b – b3 and a3
– b3 {Ans b(a2-b2)(a2+ab+b2)}
3.
Find the L.C.M. of:
a. x3 – y3
and x4 + x2y2
+ y4 {Ans (x -y) (x4+ x2y2+y4)}
b.
a3 – 1 and a4 + a2
+ 1 {Ans (a-1) (a4+a2+1)}
c.
a6 – b6 and a4
+ a2b2 + b4 (Ans a6 - b6)
4.
Find the L.C.M. of:
a.
a3 – 4a, a4 +a3
– 2a2 and 2a3 – 16 {Ans 2a2(a2-4)
(a-1) (a2+2a+4)}
b.
2x2 – 8, x2 – x - 2
and x4 – 8x {Ans
2x(x2-4) (x+1) (x2+2x+4)}
c.
4y3 -y, 2y3 – y2
– y and 8y4 + y {Ans y(4y2-1) (y-1)
(4y2-2y+1)}
d.
x3 + 1, x4 + x2 + 1
and x4 + x3 + x2 {Ans x2(x+1)(x4+x2+1)}
5.
Find the L.C.M. of:
a.
a3 + 1, a6 – 1 and a4
+ a2 + 1 (Ans a6 - 1)
b.
x4 – x, x4 + x2 + 1
and x6 –
1 {Ans x(x6-1)}
6.
Find the L.C.M. of:
a.
a3 + 2a2 – a- 2 and a3
+ a2 – 4a – 4 {Ans (a2-1)(a2-4)}
b.
x6 – 1, x4 + x3 + x2 and x3 + 2x2 + 2x + 1 {Ans x2(x6-1)}
c.
x2 + 2xy + y2
– z2, y2 + 2yz + z2 - x2 and z2
+ 2xz + x2 – y2
{Ans (x+y+z)
(x+y-z)
(y+z-x) (z+x-y)}
d.
2a3 -16, a2 + 3a + 2 and
2a2 - 8 {Ans 2(a+1) (a2-4)
(a2+2a+4)}
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