= Solution
\b[First intensity minimum.] In <Fraunhofer diffraction>, the path difference between adjacent slits is $d\sin\theta$, so the relative <wave phase> is $\delta=2\pi d\sin\theta/\lambda_1$. Summing the $N$ equal <complex amplitudes> as a <geometric series> gives the <diffraction grating> interference factor
$$
I(\theta)\propto
\left|\sum_{j=0}^{N-1}e^{ij\delta}\right|^2
=\frac{\sin^2(N\delta/2)}{\sin^2(\delta/2)}.
$$
The central maximum has $\delta=0$. Its nearest zero occurs at $N\delta/2=\pi$, where the denominator is nonzero. Thus
$$
\sin\theta_{\min}=\frac{\lambda_1}{Nd}
=\frac{640\times10^{-9}}{10(80\times10^{-6})}=8.00\times10^{-4},
\qquad
\boxed{|\theta_{\min}|=8.00\times10^{-4}\,\mathrm{rad}=0.0458^\circ.}
$$
There is a symmetric first zero on each side of the incident direction. The narrow slits are treated as having a sufficiently broad <single-slit diffraction> envelope.
\b[Wavelength resolution and the missing order specification.] Adopt the <Rayleigh criterion>: the principal maximum of one <wavelength> lies at the adjacent minimum of the other in the same nonzero <diffraction order> $m$. For $\lambda_2>\lambda_1$ and $m>0$, those directions obey
$$
d\sin\theta=m\lambda_2
=\left(m+\frac1N\right)\lambda_1.
$$
Consequently the ideal <resolving power of a diffraction grating> is $\lambda_1/\Delta\lambda=mN$, giving
$$
\boxed{\Delta\lambda_{\min,m}=\frac{\lambda_1}{mN}
=\frac{64}{m}\,\mathrm{nm}.}
$$
The usual first-order reading therefore gives \b[$64\,\mathrm{nm}$]. Zero <diffraction order> cannot separate the two <wavelengths>, because both central maxima coincide.
The paper does not specify which <diffraction order> is observed, so $64\,\mathrm{nm}$ is not an order-independent minimum. In the ideal infinitely narrow-slit model, the largest usable order for this positive wavelength separation must satisfy $(m+1/N)\lambda_1\le d$. Since $d/\lambda_1=125$, this permits $m\le124$, giving the formal all-order result
$$
\boxed{\Delta\lambda_{\min,124}=0.516\,\mathrm{nm}.}
$$
That pair meets the criterion at $\theta\simeq83.1^\circ$. Order $125$ of $\lambda_1$ is at grazing incidence and the corresponding longer-wavelength peak cannot propagate. In a real device, finite slit width, weak high-order intensity and screen acceptance may prevent such an extreme order from being used; those data are absent. Thus the specified-order expression, and $64\,\mathrm{nm}$ when first order is intended, are the defensible resolution answers under the stated criterion.
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