Solution

Calculate the value of $p$:

We know that the relation between the HCF and the LCM of any two numbers $a$ and $b$ is given by,

$\mathrm{HCF}(\mathrm{a},\mathrm{b})\times \mathrm{LCM}(\mathrm{a},\mathrm{b}) = a\times b$

It is given that, $\mathrm{LCM}(p, 12) = 24$

And, $\mathrm{HCF}(p, 12) = 4$

So, the numbers are $p$ and $12$.

Now, using the above formula,

$\mathrm{HCF}(p, 12)\times \mathrm{LCM}(p, 12) = p\times 12$

$\Rightarrow$ $4\times 24 = p\times 12$ $\left[\because \mathrm{HCF}(p, 12)=4, \mathrm{LCM}(p, 12)=24\right]$

$\Rightarrow$ $\frac{4\times 24}{12} = p$

$\Rightarrow$ $\frac{4\times 2}{1} = p$

$\Rightarrow$ $8 = p$

Or, $p = 8$

Hence, the required value of $p$ is $8$.