HCF and LCM
1. Define H.C.F.
Soln: The HCF (Highest Common Factor) of two or more numbers/expressions is the greatest number/expression that exactly divides each of the given numbers/expressions.
2. Define L.C.M.
Soln: The LCM (Lowest Common Multiple) of two or more given numbers/expressions is the smallest number/expression that is exactly divisible by each of the given numbers/expressions.
3. If P and Q are two expressions such that P divides Q without remainder, then what will be their H.C.F?
Soln: If P divides Q without remainder, then their H.C.F will be P.
4. Given any two expressions G and F if F divides G without remainder, what will be the L.C.M of G and F?
Soln: If F divides G without remainder, then their L.C.M will be G.
5. What is the H.C.F. of two expressions (x + 2) and (y + 3)?
Soln: In two expression (x + 2) and (y + 3), there is no common factor except 1. Hence F.C. F. of (x + 2) and (y + 3) is 1.
6. What is the L.C.M of two expressions (x2 + a) and (x + a)?
Soln: The L.C.M of two expressions (x2 + a) and (x + a) is (x + a)(x2 + a).
7. If H.C.F and the product of remaining factors of any two or more expressions are represented by P and Q respectively and R denotes the L.C.M of the expressions, then write the relation among P, Q and R.
Soln: The relation among P, Q and R is R = P.Q.
8. If G and L represent the HCF and LCM of two expressions P and Q respectively, write the relation among G, L, P and Q.
Soln: The relation is as follows G × L = P (x) ´ Q (x). Ans.
9. Find the LCM of :
x(x – y) and y(x + y)
Soln: Here, the first expression = x(x - y)
Second expression = y(x + y)
LCM = xy(x - y)(x + y) = xy(x2 - y2) Ans.
1. Find the HCF of :
x4 + 4 and 3x2 – 6x + 6
Soln: 1st exp. x4 + 4 = (x2)2 + 22
= (x2 + 2)2 – 4x2
= (x2 + 2)2 – (2x)2
= (x2 + 2 + 2x) (x2 + 2 – 2x)
= (x2 + 2x + 2) (x2 – 2x + 2)
2nd exp. 3x2 – 6x + 6 = 3 (x2 – 2x + 2)
\ H.C.F. = x2 – 2x + 2. Ans
2. Find the HCF of:
3x4 – 15x2 and 7x3 – 35x
Soln: Here, the first expression = 3x4 – 15x2
= 3x2(x2 – 5)
= 3 . x . x (x2 – 5)
and, second expression = 7x3 – 35x
= 7x(x2 – 5)
\ HCF = x (x2 – 5) Ans.
3. Find the HCF of:
x2 – 3x + 3y – y2 and y2 + xy – 3y
Soln: First expression = x2 – 3x + 3y – y2
= x2 – y2 – 3x + 3y
= (x – y) (x + y) – 3(x – y)
= (x – y) (x + y – 3)
Second expression = y2 + xy – 3y
= y(y + x – 3)
= y(x + y – 3)
\ H.C.F. = x + y – 3 Ans.
4. Find the HCF of :
x3 – y3 and x2 + xy + y2
Soln: Here, first expression = x3 - y3
= (x - y)(x2 + xy + y2)
Second expression = x2 + xy + y2
\ H.C.F. = x2 + xy + y2 Ans.
5. Find the HCF of:
x3 + y3 and x2 – xy + y2
Soln: Here, first expression = x3 + y3
= (x + y)(x2 - xy + y2)
Second expression = x2 - xy + y2
\ H.C.F. = x2 - xy + y2 Ans.
6. Find the HCF of:
x2 – y2 and x2 – 2xy + y2
Soln: Here, first expression = x2 - y2
= (x - y)(x + y)
Second expression = x2 - 2xy + y2
= (x - y)2
= (x - y) (x - y)
\ H.C.F. = x - y Ans.
7. Find the H.C.F ofM
x4 – 64x and x2 + 4x + 16
Soln: Here, first expression = x4 - 64x
= x(x3 - 64)
= x{x3 - (4)3}
= x (x - 4){(x2 + x .4 + (4)2 }
= x (x - 4)(x2 + 4x + 16)
Second expression = x2 + 4x + 16
\ H.C.F. = x2 + 4x + 16 Ans.
8. Find HCF of:
8m3 – 27n3; 8m2 + 12mn + 18n2
Soln: First expression= 8m3 – 27n3
= (2m)3 – (3n)3
= (2m – 3n) (4m2 + 6mn + 9n2)
Second expression = 8m2 + 12mn + 18n2
= 2(4m2 + 6mn+ 9n2)
\ L.C.M. = 2(2m – 3n) (4m2 + 6mn + 9n2)
= 2 (8m3 – 27n3) Ans.
9. Find HCF of:
4x2 – 9y2; 8x3 + 27y3
Soln: Here, first expression = 4x2 - 9y2
= (2x)2 - (3y)2
= (2x - 3y) (2x + 3y)
Second expression = 8x3 + 27y3
= (2x)3 + (3y)3
= (2x + 3y) {(2x)2 - 2x.3y + (3y)2}
= (2x + 3y) (4x2 - 6xy + 9y2)
\ H.C.F. = 2x + 3y Ans.
10. Find the H.C.F. of:
12x4 – 27x2y2 and 2x2 – xy – 3y2
Soln: Here, first expression = 12x4 - 27x2y2
= 3x2(4x2 - 9y2)
= 3x2{(2x)2 - (3y)2}
= 3x2(2x - 3y) (2x + 3y)
Second expression = 2x2 – xy – 3y2 = 2x2 – 3xy + 2xy – 3y2
= x(2x - 3y) + y(2x + 3y)
= (x - y)(2x - 3y)
\ H.C.F. = 2x - 3y Ans.
11. Find HCF of
3m3 – 27m and m3 + 27
Soln: Here, first expression = 3m3 - 27m
= 3m(m2 - 9)
= 3m{(m)2 - (3)2}
= 3x2(m - 3) (m + y)
Second expression = m3 + 27 = m3 + (3)3
= (m + 3){m2 - m.3 + (3)2}
= (m + 3)(m2 - 3m + 9)
\ H.C.F. = m + 3 Ans.
12. Find the HCF of:
x2 – y2 + x + y and x2 – xy + x
Soln: Here, first expression = x2 – y2 + x + y = (x - y) (x + y) + 1(x + y)
= (x + y) (x - y + 1)
Second expression = x2 – xy + x = x(x - y + 1)
\ H.C.F. = (x - y + 1) Ans.
13. Find the HCF of:
x2 – y2 – 2x + 1 and x2 – xy – x
Soln: Here, first expression = x2 – y2 - 2x + 1
= x2 - 2x + 1 – y2
= (x - 1)2 - y2
= (x - 1 + y) (x - 1 - y)
= (x + y - 1) (x - y - 1)
Second expression = x2 – xy -x = x(x - y - 1)
\ H.C.F. = (x - y - 1) Ans.
14. Find the LCM of:
a4 + a2 + 1, a3 + a2 + a
Soln: First expression = a4 + a2 + 1
= (a2)2 + 12 + a2
= (a2 + 1)2 – 2a2.1 + a2
= (a2 + 1)2 – a2
= (a2 + 1 – a) (a2 + 1 + a)
= (a2 – a + 1) (a2 + a + 1)
Second expression = a3 + a2 + a
= a(a2 + a + 1)
\ L.C.M. = a(a2 + a + 1) (a2 – a + 1) Ans
15. Find the LCM of:
a3 + b3, a3b – a2b2 + ab3
Soln: First expression= a3 + b3
= (a + b) (a2 - ab + b2)
Second expression = a3b – a2b2 + ab3
= ab(a2 - ab + b2)
\ L.C.M. = ab(a + b)(a2 – ab + b2) Ans
16. Find the LCM of:
p4 + 4, 2p3 – 4p2 + 4p
Soln: First expression= p4 + 4
= (p2)2 + (2)2
= (p2+ 2)2 - 2.p2.2
= (p2 + 2)2 - 4p2
= (p2 + 2)2 - (2p)2
= (p2 + 2 - 2p) (p2 + 2 + 2p)
= (p2 - 2p + 2) (p2 + 2p + 2)
Second expression = 2p3 – 4p2 + 4p
= 2p(p2 - 2p + 2)
\ L.C.M. = 2p(p2 – 2p + 2) (p2 + 2p + 2) Ans
17. Find the LCM of:
y3 – 1, y4 + y2 + 1
Soln: 1st exp. y3 – 1 = y3 – 13 = (y – 1) (y2 + y + 1)
2nd exp. y4 + y2 + 1 = (y2)2 + (1)2 + y2
= (y2 + 1)2 – 2y2 + y2
= (y2 + 1)2 – y2
= (y2 + 1 + y) (y2 + 1 – y)
= (y2 + y + 1) (y2 – y + 1)
\ L.C.M. = (y – 1) (y2 + y + 1) (y2 – y + 1) Ans.
18. Find the LCM of :
x4 + x2 + 1 and x2 – x + 1
Soln: 1st exp. = x4 + x2 + 1
= (x2)2 + (1)2 + x2
= (x2 + 1)2 – 2x2 + x2
= (x2 + 1)2 – x2
= (x2 + 1 + x) (x2 + 1 – x)
= (x2 + x + 1) (x2 – x + 1)
2nd exp. = x2 – x + 1
\ L.C.M. = (x2 + x + 1) (x2 – x + 1) = x4 + x2 + 1 Ans.
19. Find the LCM of:
a3 + 8b3 and a2 – 2ab + 4b2
Soln: Here, first expression = a3 + 8b3
= a3 + (2b)3
= (a + 2b) {a2 - a.2b + (2b)2}
= (a + 2b) (a2 - 2ab + 4b2)
Second expression = a2 – 2ab + 4b2
\ L.C.M. = (a + 2b) (a2 - 2ab + 4b2) = a3 + 8b3 Ans.
20. Find the LCM of:
(x + y)2 – 4xy and x2 + xy – 2y2
Soln: 1st exp. (x + y)2 – 4xy = x2 + 2xy + y2 – 4xy
= x2 – 2xy + y2
= (x – y)2
2nd exp. x2 + xy – 2y2 = x2 + 2xy – xy – 2y2
= x (x + 2y) – y (x + 2y)
= (x + 2y) (x – y)
\ L.C.M. = (x – y) (x – y) (x + 2y) = (x – y)2 (x + 2y) Ans.
30. Find the HCF of:
x2 + y2 + 2xy – 1, y2 – x2 + 2y + 1 and x2 – y2 + 2x + 1
Soln: 1st exp. = x2 + y2 + 2xy – 1
= (x + y)2 – 12
= (x + y + 1) (x + y – 1)
2nd exp. = y2 – x2 + 2y + 1
= (y2 + 2y + 1) – x2
= (y + 1)2 – x2
= (y + 1 + x) (y + 1 – x)
= (x + y + 1) (–x + y + 1)
3rd exp. = x2 – y2 + 2x + 1
= (x2 + 2x + 1) – y2
= (x +1)2 – y2
= (x + 1 + y) (x + 1 – y)
= (x + y + 1) (x – y + 1)
\ H.C.F. = (x + y + 1)
31. Find the HCF of:
5x2 – 125, x2 – 10x + 25 and 2x2 – 10x
Soln: 1st exp. = 5x2 – 125
= 5(x2 - 25)
= 5{x2 - (5)2}
= 5(x - 5)(x + 5)
2nd exp. = x2 – 10x + 25
= x2 – 5x - 5x + 25
= x(x - 5) – 5(x - 5)
= (x- 5 (x + 5)
3rd exp. = 2x2 – 10x
= 2x(x- 5)
\ H.C.F. = (x - 5) Ans
32. Find the HCF of:
1 + 4x + 4x2 – 16x4 and 1 + 2x – 8x3 – 16x4
Soln: 1st expression= 1 + 4x + 4x2 – 16x4
= (1 + 2x)2 – 16x4
= (1 + 2x)2 – (4x2)2
= {(1 + 2x) + 4x2} {(1 + 2x) – 4x2}
= (1 + 2x + 4x2) (1 + 2x – 4x2)
2nd expression = 1 + 2x – 8x3 – 16x4
= 1 – 8x3 + 2x – 16x4
= (1 – 8x3) + 2x(1 – 8x3)
= (1 – 8x3) (1 + 2x)
= {1 – (2x)3} (1 + 2x)
= (1 – 2x) (1 + 2x + 4x2) (1 + 2x)
\ H.C.F. = 1 + 2x + 4x2 Ans.
33. Find the HCF of:
y3 – 1, y4 + y2 + 1 and y3 + 1 + 2y2 + 2y
Soln: 1st expression = y3 – 1
= (y – 1) (y2 + y + 1)
2nd expression = y4 + y2 + 1
= (y2+ 1)2 – 2y2 + y2
= (y2 + 1)2 – y2
= (y2 + 1 + y) (y2 + 1 – y)
= (y2 + y + 1) (y2 – y + 1)
3rd expression: = y3 + 1 + 2y2 + 2y
= (y + 1) (y2 – y + 1) + 2y (y + 1)
= (y + 1) (y2 – y + 1 + 2y)
= (y + 1) (y2 + y + 1)
\ H.C.F. = (y2 + y + 1) Ans.
34. Find the LCM of:
x3 + 7x2 + 12x, x3 + 64 and 3x2 + 27x + 60
Soln: 1st expression = x3 + 7x2 + 12x
= x(x2 + 7x + 12)
= x{x2 + (4 + 3) x + 12}
= x{x2 + 4x + 3x + 12}
= x{x(x + 4) + 3(x + 4)}
= x (x + 4) (x + 3)
2nd expression= x3 + 64
= x3 + (4)3
= (x + 4) (x2 – 4x + 16)
3rd expression= 3x2 + 27x + 60
= 3(x2 + 9x + 20)
= 3{x2 + 5x + 4x + 20}
= 3{x(x + 5) + 4(x + 5)}
= 3(x +5) (x + 4)
\ L.C.M. = 3x(x + 3) (x + 4) (x + 5) (x2 – 4x + 16) Ans
35. Find the LCM of:
8x3 + 27y3, 8x3 – 27y3 and 16x4 + 36x2y2 + 81y4
1st expression = 8x3 + 27y3
= (2x)3 + (3y)3
= (2x + 3y) {(2x)2 – 2x.3y + (3y)2}
= (2x + 3y) (4x2 – 6xy + 9y2)
2nd expression = 8x3 - 27y3
= (2x)3 + (3y)3
= (2x - 3y) {(2x)2 + 2x.3y + (3y)2}
= (2x - 3y) (4x2 + 6xy + 9y2)
3rd expression = 16x4 + 36x2y2 + 81y4
= (4x2 + 9y2)2 – (6xy)2
= (4x2 + 9y2 + 6xy) (4x2 + 9y2 – 6xy)
= (4x2 + 6xy + 9y2) (4x2 – 6xy + 9y2)
\ L.C.M. = (4x2 + 6xy + 9y2) (4x2 – 6xy + 9y2) (2x - 3y) (2x + 3y)
= (8x3 + 27y3) (8x3 – 27y3) Ans
36. Find the LCM of:
x3 – 9x, x4 – 2x3 – 3x2 and x3 – 27
Soln: 1st expression = x3 – 9x
= x(x2 – 9)
= x(x + 3) (x – 3)
2nd expression = x4 – 2x3 – 3x2
= x2(x2 – 2x – 3)
= x2{x2 – 3x + x – 3}
= x2{x(x – 3) + 1(x – 3)}
= x2(x – 3) (x + 1)
3rd expression = x3 – 27
= x3 – (3)3
= (x – 3) (x2 + 3x + 9)
\ L.C.M. = x.x (x + 1) (x + 3) (x – 3) (x2 + 3x + 9)
= x2(x + 1) (x2 – 9) (x2 + 3x + 9) Ans.
37. Find the LCM of:
x4 + x2 + 1, x4 – x and 2x3 + 2x2 + 2x
Soln: 1st expression = x4 + x2 + 1
= (x2+ 1)2 – 2x2 + x2
= (x2 + 1)2 – x2
= (x2 + 1 + x) (x2 + 1 – x)
= (x2 + x + 1) (x2 – x + 1)
2nd expression = x4 – x
= x(x3 - 1)
= x(x – 1) (x2 + x + 1)
3rd expression = 2x3 + 2x2 + 2x
=2x(x2 + x + 1)
\ L.C.M. = 2x(x – 1) (x2 + x + 1) (x2 – x + 1) Ans.
38. Find the LCM of:
a2 – b2 + 2ab – c2, (a + b)2 – c2, (a + b – c)2
Soln: 1st expression = a2 – b2 + 2ab – c2
2nd expression = (a + b)2 – c2
= (a + b + c) (a + b – c)
3rd expression = (a + b – c)2
= (a + b – c) (a + b – c)
\ LCM = (a + b + c) (a + b – c) (a + b – c) (a2 – b2 + 2ab – c2)
= (a + b + c) (a + b – c)2 (a2 – b2 + 2ab – c2) Ans.
30. Find the LCM of:
ab – ac + bc – b2, bc – ab + ac – c2 and ac – bc + ab – a2
Soln: 1st expression = ab – ac + bc – b2
= ab – ac - b2 + bc
= a(b - c) - b(b - c)
= (b - c) (a - b)
2nd expression = bc – ab + ac – c2
= bc – ab - c2 + ac
= b(c - a) - c(c - a)
= (c - a) (b -c)
3rd expression = ac – bc + ab – a2
= ac – bc - a2 + ab
= c(a - b) - a(a - b)
= (a - b) (c - a)
\ LCM = (a – b) (b – c) (c – a) Ans.
31. Find the H.C.F of:
(p + r) (p – r) + q(2p + q), (q + p) (q – p) + r(2q + r) and
(r + q) (r – q) + p(2r + p)
Soln: Here, 1st expression = (p + r) (p – r) + q(2p + q)
= p2 - r2 + 2pq + q2
= (p + q)2 - r2
= (p + q - r) (p + q + r)
2nd expression = (q + p) (q – p) + r(2q + r)
= q2 - p2 + 2qr + r2
= (q + r)2 - p2
= (q + r - p) (q + r + p)
3rd expression = (r + q) (r – q) + p(2p + p)
= r2 - q2 + 2pq + p2
= (q + r)2 - p2
= (q + r - p) (q + r + p)
\ H.C.F. = (p + q + r) Ans.
32. Find the L.C.M of:
a(a + c) – b(b + c), b(a + b) – c(c + a) and c(b + c) – a(a + b)
Soln: First exp. = a(a + c) – b(b + c)
= a2 + ac - b2 - bc
= a2 - b2 + ac - bc
= (a + b) (a - b) + c(a - b)
= (a - b) (a + b + c)
Second exp. = b(a + b) – c(c + a)
= ab + b2 - c2 - ac
= b2 - c2 + ab - ac
= (b + c) (b - c) + a(b - c)
= (b - c) (b + c + a)
Third exp. = c(b + c) – a(a + b)
= bc + c2 - a2 - ab
= c2 - a2 + bc - ab
= (c + a) (c - a) + b(c - a)
= (c - a) (c + a + b)
\ L.C.M. = (a + b + c) (a – b)(b – c)(c – a) = (a – b)(b – c)(c – a)(a + b + c)
33. L3 + M3 = P3 + Q3.If H is the HCF and L, the LCM of two quantities P and Q, and if L + M = P + Q, then prove that: L3 + M3 = P3 + Q3
Soln: Here, H.C.F. = H, L.C.M. = L, L + M = P + Q
We know that, product of H.C.F. and L.C.M. = product of two expressions
or, HL = PQ
Now, we have
L + M = P + Q
Cubing on both sides,
(L + M)3 = (P + Q)3
or, L3 + M3 + 3LM (L + M) = P3 + Q3 + 3PQ (P + Q)
or, L3 + M3 + 3PQ (P + Q) = P3 + Q3 + 3PQ (P + Q)
\ L3 + M3 = P3 + Q3. Proved.
34. Find the L.C.M. of):
1 – x2, 1 – 6x + 5x2, 1 – 4x – 20x3 – 25x4
Soln: 1st exp. = 1 – x2
= (1 + x) (1 – x)
2nd exp. = 1 – 6x + 5x2
= 1 – 5x – x + 5x2
= (1 – 5x) – x (1 – 5x)
= (1 – 5x) (1 – x)
3rd exp. = 1 – 4x – 20x3 – 25x4
= (1 – 25x4) – 4x – 20x3
= (1 + 5x2) (1 – 5x2) – 4x (1 + 5x2)
= (1 + 5x2) (1 – 5x2 – 4x)
= (1 + 5x2) (1 – 4x – 5x2)
= (1 + 5x2) (1 – 5x + x – 5x2)
= (1 + 5x2) {(1 – 5x) + x (1 – 5x)}
= (1 +5x2) (1 – 5x) (1 + x)
\ L.C.M. = (1 + x) (1 – x) (1 – 5x) (1 + 5x2)
= (1 – x2) (1 – 5x) (1 + 5x2) Ans.
35. Find the LCM of:
m2 + m + 1, m3 – 1, m6 – 1
Soln: 1st exp. = m2 + m + 1
2nd exp. = m3 – 1
= m3 – 13
= (m – 1) (m2 + m + 1)
3rd exp. = m6 – 1
= (m3)2 – 12
= (m3 + 1) (m3 – 1)
= (m3 + 1) (m – 1) (m2 + m + 1)
\ L.C.M. = (m – 1) (m2 + m + 1) (m3 + 1) = (m3 – 1) (m3 + 1) = (m6 – 1) Ans.
36. Find the HCF of) :
a4 – 3a3 + 3a – 1, a6 – 1 and a3 – a
Soln: Here,
1st expn.= a4 – 3a3 + 3a – 1
= a4 – a3 – 2a3 + 2a2 – 2a2 + 2a + a – 1
= a3(a – 1) – 2a2(a – 1) – 2a(a – 1) + 1(a – 1)
= (a – 1) (a3 – 2a2 – 2a + 1)
= (a – 1) {a3 + a2 – 3a2 – 3a + a + 1}
= (a – 1) {a2(a + 1) – 3a(a + 1) + 1(a + 1)}
= (a – 1) (a + 1) (a2 – 3a + 1)
= (a – 1) (a + 1) (a2 – 3a+ 1)
2nd expn. = a6 – 1
= (a3)2 – (1)2
= (a3 – 1) (a3 + 1)
= (a – 1) (a2 + a + 1) (a + 1) (a2 – a + 1)
3rd expn. = a3 – a
= a(a2 – 1)
= a(a – 1) (a + 1)
\ H.C.F. = (a – 1) (a + 1) = a2 – 1 Ans.
37. (Find the LCM of):
9x4 – 28x2 + 3, 27x4 – 12x2 + 1 and x4 – 6x2 + 9
Soln: 1st expression = 9x4 – 28x2 + 3
= 9x4 – 27x2 – x2 + 3
= 9x2 (x2 – 3) – 1(x2 – 3)
= (x2 – 3) (9x2 – 1)
= (x2 – 3) {(3x)2 – (1)2}
= (x2 – 3) (3x – 1) (3x + 1)
2nd expression = 27x4 – 12x2 + 1
= 27x4 – 9x2 – 3x2 + 1
= 9x2(3x2 – 1) – 1(3x2 – 1)
= (3x2 – 1) (9x2 – 1)
= (3x2 – 1) {(3x)2 – (1)2}
= (3x2 – 1) (3x – 1) (3x + 1)
3rd expression = x4 – 6x2 + 9
= (x2)2 – 2.x2.3 + (3)2
= (x2 – 3)2
LCM = (x2 – 3) (3x – 1) (3x + 1) (3x2 – 1) (x2 – 3)
= (9x2 – 1) (3x2 – 1) (x2 – 3)2 Ans.
38. (Find the HCF of):
x3 – 64y3, x2 – 6xy + 8y2 and x2 – 16y2
Soln: First expression = x3 – 64y3 = x3 – (4y)3
= (x – 4y) (x2 + x × 4y + 16y2)
= (x – 4y) (x2 + 4xy + 16y2)
Second expression = x2 – 6xy + 8y2
= x2 – 4xy – 2xy + 8y2
= x (x – 4y) – 2y (x – 4y)
= (x – 2y) (x – 4y)
Third expression = x2 – 16y2 = x2 – (4y)2
= (x + 4y) (x – 4y)
\ H.C.F. = (x – 4y) Ans.
39. (Find the LCM of):
x2 – 7xy + 12y2, x2 + 2xy – 15y2 and x2 + xy – 20y2
Soln: 1st expression = x2 – 7xy + 12y2
= x2 – 4xy – 3xy + 12y2
= x(x – 4y) – 3y(x – 4y)
= (x – 4y) (x – 3y)
2nd expression = x2 + 2xy – 15y2
= x2 + 5xy – 3xy – 15y2
= x(x + 5y) – 3y(x + 5y)
= (x + 5y) (x – 3y)
3rd expression = x2 + xy – 20y2
= x2 + 5xy – 4xy – 20y2
= x(x + 5y) – 4y(x + 5y)
= (x + 5y) (x – 4y)
\ L.C.M. = (x – 3y) (x – 4y) (x + 5y) Ans.
40. Find the H.C.F. of) :
b2 (b2 + 4bc + 4c2), b5 + 8b2c3 and 3b4 + b3c – 10b2c2
Soln: Here,
1st exp. =b2 (b2 + 4bc + 4c2)
= b2 (b + 2c)2
2nd exp. = b5 + 8b2c3
= b2 (b3 + 8c3)
= b2 {b3 + (2c)3}
= b2 (b + 2c) (b2 – 2bc + 4c2)
3rd exp. = 3b4 + b3c – 10b2c2
= b2 (3b2 + bc – 10c2)
= b2 {3b2 + 6bc – 5bc – 10c2}
= b2 {3b (b + 2c) – 5c(b + 2c)}
= b2 (b + 2c) (3b – 5c)
\ HCF = b2 (b + 2c) Ans.
41. Find the H.C.F. of) :
p2 + 4pq + 4q2, p4 + 8pq3 and 3p4 – 10p2q2 + p3q
Soln: 1st exp.= p2 + 4pq + 4q2
= (p + 2q)2
= (p + 2q) (p + 2q)
2nd exp.=p4 + 8pq3
= p(p3 + 8q3)
= p{p3 + (2q)3}
= p(p + 2q) (p2 – 2pq + 4q2)
3rd exp. = 3p4 – 10p2q2 + p3q
= (3p4 + p3q – 10p2q2)
= p2(3p2 + pq – 10q2)
= p2(3p2 + 6pq – 5pq – 10q2)
= p2 {3p(p + 2q) – 5q(p + 2q)}
= p2(p + 2q) (3p – 5q)
\ HCF = (p + 2q) Ans.
42. (Find the H.C.F. of) :
v2(v2 + 4vw + 4w2), v5 + 8v2w3 and 3v4 + v3w – 10v2w2
Soln: Here,
First expression = v2(v2 + 4vw + 4w2)
= v2(v2 + 2vw + 2vw + 4w2)
= v2{v(v + 2w) + 2w(v + 2w)}
= v2(v + 2w) (v + 2w)
Second expression = v5 + 8v2w3
= v2(v3 + 8w3) = v2{v3 + (2w)3}
= v2(v + 2w) (v2 – 2vw + 4w2)
Third expression = 3v4 + v4w – 10v2w2
= v2(3v2 + vw – 10w2)
= v2(3v2 + 6vw – 5vw – 10w2)
= v2{3v(v + 2w) – 5w(v + 2w)}
= v2(v + 2w) (3v – 5w)
Hence, HCF = v2 (v + 2w) Ans.
43. (Find the H.C.F. of):
a2 + 4ab + 4b2, a4 + 8ab3 and 3a4 – 10a2b2 + a3b
Soln: Here,
First expression = a2 + 4ab + 4b2
= a2 + 2ab + 2ab + 4b2
= a(a + 2b) + 2b(a + 2b)
= (a + 2b) (a + 2b)
Second expression = a4 + 8ab3
= a(a3 + 8b3)
= a{a3 + (2b)3}
= a(a + 2b) {a2 – a.2b + (2b)2}
= a(a + 2b) (a2 – 2ab + 4b2)
Third expression = 3a4 – 10a2b2 + a3b
= a2(3a2 + ab – 10b2)
= a2(3a2 + 6ab – 5ab – 10b2)
= a2 {3a(a + 2b) – 5b(a + 2b)}
= a2(a + 2b) (3a – 5b)
Hence, HCF = (a + 2b) Ans.
44. (Find the H.C.F. of) :
x3 + 4x2y + 4xy2, x4 + 8xy3 and 3x4 + x3y – 10x2y2
Soln: Here,
First expression = x3 + 4x2y + 4xy2
= x(x2 + 4xy + 4y2)
= x (x2 + 2xy + 2xy + 4y2)
= x{x(x + 2y) + 2y(x + 2y)}
= x(x + 2y) (x + 2y)
Second expression = x4 + 8xy3
= x(x3 + 8y3)
= x{x3 + (2y)3}
= x(x + 2y) {x2 – x.2y + (2y2})
= x(x + 2y) (x2 – 2xy + 4y2)
Third expression = 3x4 + x3y – 10x2y2
= x2(3x2 + xy – 10y2)
= x2(3x2 + 6xy – 5xy – 10y2)
= x2{3x(x + 2y) – 5y(x + 2y)}
= x2(x + 2y) (3x – 5y)
Hence, H.C.F. = x(x + 2y) Ans.
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