The reciprocal of the sum of the reciprocals of
5
6
​
and
7
3
​
is:

First, find the reciprocals of the given numbers
5
6
​
and
7
3
​
:

Reciprocal of
5
6
​
=
6
5
​


Reciprocal of
7
3
​
=
3
7
​


Now, find the sum of these reciprocals:

Sum=
6
5
​
+
3
7
​

To add these fractions, find a common denominator, which is 6:

Sum=
6
5
​
+
3×2
7×2
​
=
6
5
​
+
6
14
​
=
6
19
​

The question asks for the reciprocal of this sum:

Reciprocal of the sum=
19
6
​

Let's look at the given options to match the fraction format. We can multiply the numerator and denominator by 3:

19×3
6×3
​
=
57
18
​

Thus, it matches option A (and C, which has a duplicated label in the original set; option A is
57
18
​
).