Show that for positive integers a and b a b a b 3aSolution

Show that for positive integers a and b, (a, b) = (a, b + 3a).

Assuming that the notation (a,b) implies gcd of a and b

Let, g=gcd(a,b)

Case 1. g=1

So a and b have no common factors

Let, gcd(a,b+3a) =k>1

So,k|a,k|b+3a

So, a=mk

b+3a=rk=b+3mk

b=(r-3m)k

b is a positive integer

Hence, k|b and k|a

But a and b have no common factor

So a contradiction

Hence

gcd(a,b+3a)=1

Case 2. g>1

Let ,gcd(a,b+3a)=k

So, a=mk,b+3a=nk=b+3mk

b=(n-3m)k,a=mk

But g is gcd of a,b and hence, k|g is k<g

But, a=rg,b=sg

So,b+3a=sg+3rg=(s+3r)g

Hence, g |a, g|b+3a

Hence, g|k

Hence g=k

Hence proved