The sum of two numbers is $36$ and their HCF is $4$. How many pairs of these numbers are possible?

Let the two numbers be $4a$ and $4b$, where $a$ and $b$ are co-prime integers (i.e., $\text{HCF}(a,b)=1$).
According to the question, their sum is $36$:

$$4a+4b=36$$

$$4(a+b)=36$$

$$a+b=9$$

Now, we look for pairs of co-prime numbers $(a,b)$ whose sum is $9$:

1. $(1,8)\Rightarrow \text{HCF}(1,8)=1$ (Valid)
2. $(2,7)\Rightarrow \text{HCF}(2,7)=1$ (Valid)
3. $(3,6)\Rightarrow \text{HCF}(3,6)=3\ne 1$ (Invalid)
4. $(4,5)\Rightarrow \text{HCF}(4,5)=1$ (Valid)

Thus, there are exactly $3$ possible pairs.