GCD

Note that, if a,b are two integers such that, lcm is their least common multiple and gcd is their greatest common divisor then, $lcm\quad \times \quad gcd\quad =\quad a\quad \times \quad b$

and , a,b can also be written as a=gx * [here,g stands for gcd] and * b=gy

Now , we know that their lcm = 7g

hence, from the points above we can say ,

$7g\quad \times \quad g\quad =\quad gx\quad \times \quad gy$

or, $7\quad =\quad x\quad \times \quad y$

as x and y must be integers , either one of them must be 1 and the other one should be 7

now we also know that a+b=56 so,

gx+gy=56

g(x+y)=56

Hence g=7 [as x+y must be 8]

$gcd(a,b)*lcm[a,b]=7k$

$\Rightarrow$ There are 7 possible pairs.

$\Rightarrow$ $(1,7),(2,14),(3,21),(4,28),(5,35),(6,42),(7,49)$

$\Rightarrow$ From the pairs $7+49=56$ satisfies the condition.
Thus the answer is $gcd(7,49)=$ $\boxed{7}$

Product of lcm and hcf is equal to the product of numbers. hcf cannot be greater than any of the two numbers. The numbers will be multiples of 7.
Cases will be 7,49 14,42 21,35 and 28,28. Checking hcf lcm for the pairs 7 is the answer.