# The sum of the greatest common divisors

What are the values for the positive numbers $a,b$ and $c$ can take the expression $$(a^2,b^2)+(a,bc)+(b,ca)+(c,ab)?$$ (Here $(u,v)=\gcd(u,v)$ - the greatest common divisor for $u\in \mathbb N, v\in \mathbb N$)

I have no clue how to start. Any kind of help will be appreciated.

i.e:

Let $A=\gcd(a^2,b^2)+\gcd(a,bc)+\gcd(b,ca)+\gcd(c,ab)$, where $a,b,c -$ positive integers. Find $A$

Addition:

It is understood that $A\ge4.$

Let $\gcd(a,b)=\gcd(a,c)=\gcd(c,b)=1$, then $A=4$

But $A\not=5$ for any $a,b,c$

If $a=c=2; b=1$ then $A=(a^2,b^2)+(a,bc)+(b,ca)+(c,ab)=1+2+1+2=6$

But $A\not=7$ for any $a,b,c$

• Can you write your question properly. – sayan kundu Apr 17 '16 at 18:28
• for this I need help too. my English is bad – Roman83 Apr 17 '16 at 18:32
• Don't worry.Just write what you understand.just write it and don't think about the mistake.It's normal. – sayan kundu Apr 17 '16 at 18:36
• Ok. Let $A=\gcd(a^2,b^2)+\gcd(a,bc)+\gcd(b,ca)+\gcd(c,ab)$, where $a,b,c -$ positive integers. Find $A$ – Roman83 Apr 17 '16 at 18:40
• You have to say what is $(a,c)$ and $(b,c)$ – N.S.JOHN Apr 18 '16 at 13:43