Let $a$ and $b$ be positive integers with no common factors. Then
$a)$ $a+b$ and $a-b$ have no common factor other than $3$ , whatever be $a$ and $b$
$b)$ $a+b$ and $a-b$ have no common factor greater than 2,whatever be $a$ and $b$
$c)$ $a+b$ and $a-b$ have a common factor, whatever be $a$ and $b$
$d)$ none of the foregoing statements is correct.
My approach : I am not proficient in number theory , so my approach till now was pretty lame . I basically included trying to check for the numbers. I first tried to check using consecutive numbers since they are always co primes. However I realised in this case I would be missing out those kind of co-primes both of whose constituents are odd numbers. This is where I hit the roadblock. ( My background is a degree in Electrical Engineering , though with no formal course/training in Number Theory.)
Please tell me the correct approach to solve the question.