The speed of electrons in a scanning electron microscope is $$1 \times 10^7$$ ms$$^{-1}$$. If the protons having the same speed are used instead of electrons, then the resolving power of scanning proton microscope will be changed by a factor of:

The resolving power of a microscope depends on the wavelength of the probe particle — a shorter wavelength allows finer details to be resolved. Specifically, resolving power is inversely proportional to the de Broglie wavelength $$\lambda$$.

The de Broglie wavelength of a particle with mass $$m$$ and speed $$v$$ is given by $$\lambda = \frac{h}{mv}$$, where $$h$$ is Planck's constant. For a given speed $$v$$, a heavier particle will have a shorter wavelength.

For the electron: $$\lambda_e = \frac{h}{m_e v}$$. For the proton: $$\lambda_p = \frac{h}{m_p v}$$. Since both travel at the same speed, the ratio of wavelengths is $$\frac{\lambda_e}{\lambda_p} = \frac{m_p}{m_e} \approx 1837$$.

This means the proton's de Broglie wavelength is $$1837$$ times smaller than the electron's. Since resolving power is inversely proportional to wavelength, the resolving power of the proton microscope is $$1837$$ times greater than that of the electron microscope.

Therefore, the resolving power changes by a factor of $$1837$$.