We study the following SU(N+1) Toda systemΔui+∑j=1Naijeuj=4π∑k=0mni,kδpk onEτ,i=1,⋯,N≥2, where Eτ:=C/(Z+Zτ) with Imτ>0 is a flat torus, (aij) is the standard Cartan matrix of SU(N+1), δpk is the Dirac measure at pk, and ni,k∈N satisfy natural non-critical conditions. It is known from [3,25] that solutions exist. In this paper we show that this system has at most∏k=0m∏j=1N∏i=1j∑l=ij(nl,k+1)(N+1)(∏j=2N(j!))m+1∈N solutions. We have several examples to show that this upper bound should be sharp. As a corollary we obtain the uniqueness of the solution for the case m=0 and (n1,0,⋯,nN−1,0,nN,0)=(0,⋯,0,1). Some results about the symmetry of solutions are also established. As applications, we obtain sharp non-existence and existence results for the SU(N+1) Toda system in the plane with four singular sources.