<s>[INST] If $a$ , $b$, $c$ are positive integers satisfying the following $(a^2 +2)(b^2+3)(c^2+4)=2014$ What is the value of $ a+b+c $ If $a$ , $b$, $c$ are positive integers satisfying the following
$(a^2 +2)(b^2+3)(c^2+4)=2014$
What is the value  of $ a+b+c $
I need  details  because  i don't  know  how to solve simlar  problems ?
Thank you for your help. 
 [/INST] let us consider following program
    z=input('enter your number : ');
  string='';
    for ii=2:z
       s=0;
       while z/ii==floor(z/ii) % check if  z is divisible by ii
           z=z/ii;
           s=s+1;
       end
       if s>0
                str =[num2str(ii) '^' num2str(s) ];

                   string=strcat(string,str);
                   string= strcat(string,'*'); 
             % If z = 1, no more divisions are necessary, 
            % thus breaks the loop and quits
            if z == 1
                break
            end
        end

    end
    string=string(1:end-1);% remove last sign of multiplicaiton
fprintf('prime factorization is %s\n',string);

we would have  
>> integer_factorization
enter your number : 2014
prime factorization is 2^1*19^1*53^1

now 
 x=2
y=19
z=53

$a^2+2=2$ 
$a^2=0$ 
which means $a=0$
$c^2+4=53$ 
$c^2=53-4=49$
which  means  that $c=7$  or $c=-7$ depend  we have positive  integers or integers and  also
$b^2+3=19$
we have $ b=4 $ or $b-4$
</s>