Question

if g.c.d (a, b)=8 , l.c.m (a, b) 64 and a>b than a=??​

Answer 1

Step-by-step explanation:

It is given that GCD(a,b)=1

Let GCD(a−b,a+b)=d

⇒d divides a−b and a+b

there exists integers m and n such that

a+b=m×d ..........(1)

and a−b=n×d ..........(2)

Upon adding and subtracting equation (1) and (2) we get

2a=(m+n)×d ..........(3)

and 2b=(m−n)×d ..........(4)

Since, GCD(a,b)=1(given)

∴2×GCD(a,b)=2

∴GCD(2a,2b)=2 since GCD(ka,kb)=kGCD(a,b)

Upon substituting value of 2a and 2b from equations (3) and (4) we get

∴gcd((m+n)×d,(m−n)×d)=2

∴d×gcd((m+n),(m−n))=2

∴d× some integer=2

∴d divides 2

∴d≤2 if x divides y, then ∣x∣≤∣y∣

∴d=1 or 2 since, gcd is always a positive integer.