The HCF and LCM of two numbers are $2$ and $72$ respectively. The bigger number is $2$ more than twice the smaller number. Find the smaller number.

Let the smaller number be $x$.
Then, according to the question, the bigger number $=2x+2$.

We know the formula:

$$\text{Product of two numbers}=\text{HCF}\times \text{LCM}$$

$$x(2x+2)=2\times 72$$

$$2x^2+2x=144$$

Divide the entire equation by $2$:

$$x^2+x=72$$

$$x^2+x-72=0$$

Factorize the quadratic equation:

$$x^2+9x-8x-72=0$$

$$x(x+9)-8(x+9)=0$$

$$(x-8)(x+9)=0$$

Since the numbers must be positive integers, $x=8$.
Therefore, the smaller number is $8$.