Question

If ax + by=1; a,b,x,y are integers. which of the following is not true:

a) (a.b)=1

b) (x,y)=1

c) (a,y)=1

d) (b,y)=1

Answer 1

Given: ax+by=1 , where a, b, x, y are integers.

First of all, take

gcd(a,b) that divides both a and b

that is , ax+by=gcd(a,b)

∴dividing both sides of the equation by gcd of (a, b), we get,

ax+by=1

Now, if a'x+b'y=1, then

gcd(x,y)=1

where gcd(x,y) is the smallest +ve integer which can be written in the form of

cx+dy ; c,d are integers.

Also, gcd(a, b) is the least positive linear combination, which must be 1.

As, we know that, 1 is a linear combination of a and b.

Thus gcd(a,b) = 1.

option d) is not true.