The sum of two positive integers is $2310$ and $11$ is their g.c.d. How many pairs of such numbers are possible?
Since $11$ has been given as the g.c.d of the two numbers, we can write the numbers as $11a$ and $11b$ such that $(a,b)=1$
As per the question's statement $$11a +11b=2310$$ $$a+b=210$$ The question has been reduced till the point where it needs (as per this approach) brute force calculation (making the pairs and checking whether they are co-primes). But this question was given to me by a friend who is preparing for an exam where questions are to be solved (usually) in around a minute. Does there exist a way in which this problem can be solved with a better approach?