Calculating Current across 3 Resistors in Parallel, each of different value, with unknown voltage but a total Current of I=5A

This problem is deceivingly simple. You know the total current and the individual resistances. But you don't know the total resistance or the individual currents or the Voltage. Obtaining the Voltage is key to solving for the current across each resistor. And to get the Voltage, we must solve for the overall resistance.

Req = 4/3Ω => V = 6.66V => I3 = 0.56A