Given:
The sum of two numbers is 36.
The highest common factor (H.C.F) of the two numbers is 3.
The least common multiple (L.C.M) of the two numbers is 105.

To find:
The sum of the reciprocals of the two numbers.

Solution:

Let the two numbers be x and y.

Step 1: Finding the numbers
Since the sum of the two numbers is 36, we can write the equation:
x + y = 36 ...(1)

Since the H.C.F of the two numbers is 3, both x and y are multiples of 3. Let's express x and y in terms of 3:
x = 3a and y = 3b, where a and b are positive integers.

Substituting these values in equation (1), we get:
3a + 3b = 36
Dividing both sides by 3, we get:
a + b = 12 ...(2)

Now, to find the values of a and b, we need more information. Let's move on to step 2.

Step 2: Finding the L.C.M
The L.C.M of two numbers can be found using the formula:
L.C.M = (x * y) / H.C.F

Substituting the given values, we have:
105 = (x * y) / 3

Multiplying both sides by 3, we get:
3 * 105 = x * y
315 = x * y ...(3)

Step 3: Solving equations (2) and (3)
Now, we have two equations:
a + b = 12 ...(2)
x * y = 315 ...(3)

We can solve these equations to find the values of a, b, x, and y.

The pairs of positive integers (a, b) that satisfy equation (2) and have a product of 315 (equation 3) are: (5, 7), (7, 5), (3, 15), (15, 3).

Substituting the values of a and b in equation (2), we get:
a + b = 12
5 + 7 = 12 (or) 7 + 5 = 12
a = 5, b = 7 (or) a = 7, b = 5

So, we have two pairs of values for x and y:
x = 3a = 3 * 5 = 15, y = 3b = 3 * 7 = 21
x = 3a = 3 * 7 = 21, y = 3b = 3 * 5 = 15

Step 4: Finding the sum of the reciprocals
The reciprocals of x and y are 1/x and 1/y, respectively.

For the first pair (x = 15, y = 21):
Sum of the reciprocals = 1/15 + 1/21

Taking the L.C.M of 15 and 21, we get:
L.C.M = 105

Converting the fractions with a common denominator:
Sum of the reciproc