We have studied properties of sustained gamma fluxes having quantum energies of >100 MeV at different stages of flares with 1-min temporal resolution (Fermi/LAT). The most probable process of emergence of high-energy gamma-quanta during the impulsive phase of flares (6 events) has been confirmed. Acceleration of particles, produced by flare energy release (at dissipation of current sheet), occurs when they interact with a shock front of a coronal mass ejection (CME), which develops in the same active region at the same time. Nuclear interactions of accelerated protons (>500 MeV) with plasma ions lead further to the emergence of high-energy gamma-quanta. We have established that the interaction between a flare flux and a high-speed CME during the flare impulsive phase occurs within fairly limited periods — from 2 to 16 min. In the events considered, we have found a direct connection between maximum gamma flux F max (γ > 100 MeV) and CME velocity. High maximum values of gamma fluxes are typical of the flare impulsive phase: 3.5·10⁻⁴ cm⁻²s⁻¹ ≤ F max (γ > 100 MeV) ≤ 1.3·10⁻² cm⁻² s⁻¹. At the same time, the value F max (γ > 100 MeV) = 0.013 cm⁻²s⁻¹ was the highest for the events observed by Fermi/LAT from 2008 to 2017. During the development of CMEs moving with a supersonic speed, shock waves are formed which are the major power source of accelerated particles during the main phase of gradual flares. In some cases, however, the impact of shock waves on particle acceleration is the greatest in the short impulsive phase. To reveal parameters most effectively influencing the generation of high-energy gamma-ray emission, we have compared 17 flare events. The most significant parameter proved to be the time interval of joint action of flare process and CME shocks. We have established that during simultaneous development of flare process and CME attendant on the flare, the most efficient particle acceleration occurs which gives rise to maximum fluxes of high-energy gamma-quanta.