 Solution

With the negative exponential convention of the specified quantum Fourier transform, the amplitude of computational outcome $k$ isIf $k=m$, every summand is one. Otherwise this finite geometric series has ratio $q\ne 1$ with $q^N=1$, so it equals $(1-q^N)/(1-q)=0$. ThusThe Born rule gives outcome $m$ with probability one when performing a quantum measurement in the computational basis. Recover the promised phase as ${\varphi }_m=2\pi m/N$ modulo $2\pi$. Using the opposite Fourier sign would instead return the label $-m\mathrm{mod}N$, so the sign convention matters.