Determine Frequency Resolution samples Can someone tell me if the way I solved this problem is right? 

Suppose I have a signal $X_a(t)= 0.5cos(700πt)+0.6cos(720πt)+0.1cos(780πt)$ where $F_s=8000Hz$

I want to determine the minimum samples which must be available in order the frequency resolution of DFT be ok.

I did this:


Firstly calculate each frequency of the signal:
$F_1=350Hz$, $F_2=360Hz$, $F_3=390Hz$
Calculate the differences: $F_2$-$F_1$=$10Hz$ ,$F_3$-$F_2$= $30Hz$ , $F_3$-$F_1$= $40Hz$
$ΔF_{min}$=$10Hz$
$ΔF$ = $ \frac{f_s}{N} $, so $N\geq\frac{8000Hz}{10}$=800 samples


Can someone tell me if my solution is right and explain why we choose the minimum difference

Thanks
It's usually considered difficult to resolve a pair of adjacent frequencies unless there is dip between the two FFT peaks.  Depending on the window (rectangular or otherwise) and how much of a dip you require between the two peaks to consider them resolved (above some floor), and whether or what kind of result interpolation you use, you will need from a bit less than double to triple or more of your suggested 800 FFT result bins.

Of course in a-priori zero noise, your signal of 3 pure sinusoids has 9 free parameters, so you might be able to determine your 3 input frequencies with perhaps as few as 9 or a bit more non-aliased samples.  In any case, much less than 800.

The resolution of a single isolated frequency peak above a low enough noise floor to the nearest 10Hz can likely be done using far fewer than 800 samples by using interpolation methods on the FFT result, depending on the SNR.