Scaling FFT output by number of points in FFT
The 1/N scaling factor is almost arbitrary placed.  An unscaled FFT followed by an unscaled IFFT using exactly the same complex exponential twiddle factors multiplies the input vector by scaler  N.  In order to get back the original waveform after an IFFT(FFT())round trip (thus making them inverse functions), some FFT/IFFT implementation pairs scale the FFT by 1/N, some scale the IFFT by 1/N, some scale both by 1/sqrt(N).