Three numbers which are co-primes of each other such that the product of the first $2$ is $551$ and that of the last $2$ is 1073. Find the $3$ numbers

Question

If we take a,b, and c as the three numbers then, I know the answer is got by using the fact that b will be the common factor of $551$ and $1073$. But what I don't understand is why is b taken as the gcd of $551$ and $1073$ as it can easily be just any of the common factors of those two numbers.

Answer 1

Assuming $a,b,c$ are all positive.

$$ab=551$$ $$bc=1073$$

$b$ clearly is a common divisor.

Suppose it is not the greatest common divisor, then $a$ and $c$ would share some common factors that are bigger than $1$ which contradicts to the fact that they are coprime.

Answer 2

If $a,b,c $ are the numbers then $ab $ and $bc $ are the products. Those have $b $ as a common factor. But $a$ and $c $ are relatively prime and have no factors in common. So $ab$ and $bc$ can't have any factors in common that aren't a factor of $b $.

So $b$ is the greatest common factor of $ab $ and $bc $. So we can find $b $. Just divide $ab$ and $bc $ by $b$ to get $a $ and $c $.