Question

For any integer a show that (2a+1,9a+4)=1​

Answer 1

Answer:

Let

d=gcd(2a+1,9a+4)

Then

d|2a+1

and

d|9a+4

2a+1=db

and

9a+4=dc

a=db−12

and

a=dc−49

Equating both equations :

9db−9=2dc−8

d(9b−2c)=1

9b−2c=1d

Now, since b and c are integers, therefore 1d is an integer, i.e d divides 1 and therefore

gcd(a,b)=d=1

.