1. Threshold condition for photodiode

A photodiode starts conducting when the incident photon energy is at least equal to its band gap energy.

So, at threshold wavelength $\lambda_0 = 660\,\text{nm}$,

$E_g = \frac{hc}{\lambda_0}$

2. Substitute the given values

Given:

$h = 6.6 \times 10^{-34}\,\text{J s}$ $c = 3 \times 10^8\,\text{m/s}$ $\lambda_0 = 660\,\text{nm} = 660 \times 10^{-9}\,\text{m}$

Thus,

$E_g = \frac{6.6 \times 10^{-34} \times 3 \times 10^8}{660 \times 10^{-9}}$

$E_g = \frac{19.8 \times 10^{-26}}{660 \times 10^{-9}}$

$E_g = 3 \times 10^{-19}\,\text{J}$

3. Convert joules into electron volts

Since

$1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J}$

Therefore,

$E_g = \frac{3 \times 10^{-19}}{1.6 \times 10^{-19}}\,\text{eV} = 1.875\,\text{eV}$

4. Relate to the given form

It is given that band gap is

$E_g = \left(\frac{X}{8}\right)\,\text{eV}$

So,

$\frac{X}{8} = 1.875 = \frac{15}{8}$

Hence,

$X = 15$

5. Check options

The correct option is:

$\boxed{\text{C: }15}$