Q. 43 O5.0( 1 Vote )

# If g. c. d (a, b) = 1, then g. c. d (a — b, a b) = ………A. 1 or 2B. a or bC. a + b or a— bD. 4

Given g.c.d. (a,b) = 1

From Euclid’s algorithm we can say that if b|a, then g.c.d(a,b) = b

Let g. c. d (a—b, a + b) = k

Therefore we can say that k is a factor of both (a–b) and (a + b).

We can write a–b = rk for some r N

And also a + b = sk for some sN

Now, (a + b) + (a–b) = rk + sk

a + a + b–b = k(r + s)

2a = k(r + s)………..eq(1)

(a + b) – (a – b) = rk – sk

a + b–a + b = k(r–s)

2b = k(r–s)………..eq(2)

Also, g. c. d (a, b) = 1

Therefore, 2 × g. c. d (a, b) = 2 × 1

g. c. d (2a, 2b) = 2

g. c. d [(r + s)k, (r – s)k] = 2 (from eq (1) and eq (2))

k × g. c. d(r + s, r – s) = 2

= 2 × 1

So, k × g. c. d(r + s, r – s) = 2

k × g. c. d(r + s, r – s) = 2 × 1 = 2 × g.c.d(a,b)

By comparing we get k = 2

We know that 1 and 2 are co–prime numbers.

Similarly, we get k = 1

So, g. c. d (a—b, a + b) = k = 2

or g. c. d (a—b, a + b) = k = 1

g. c. d (a—b, a + b) = 1 or 2

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Interactive Quiz:Euclid's Division Lemma44 mins
Fundamental Theorem of Arithmetic-238 mins
Champ Quiz | Fundamental Principle Of Arithmetic41 mins
Fundamental Theorem of Arithmetic- 143 mins
NCERT | Imp. Qs. on Rational and Irrational Numbers44 mins
Euclids Division Lemma49 mins
Quiz | Imp Qs on Real Numbers37 mins
Interactive Quiz - HCF and LCM32 mins
Application of Euclids Division Lemma50 mins
Relation Between LCM , HCF and Numbers46 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses