Greetings in that Divine and Most Precious Name of Our Lord and Savior Jesus Christ
He was 100% God and 100% man. So, His divinity could not be tempted, but His humanity could.
And his divinity did not nullify, override, silence or absorb his humanity.
These do not really address the idea, further they create an image of a kind of schizophrenic Jesus Christ whose Person was splintered or divided between two thought processes, the Divine and Human. Jesus Christ the Incarnate Word was God-Man, but through the Union there is only ONE unified Nature, not a split personality. This kind of theological reasoning is precisely why we Orientals reject Chalcedon, because we feel it creates this schizophrenic Jesus Christ. When the Divine and Human natures were fully unified into the SINGLE composite nature in the Incarnation they cooperated together as one personality, one mind, one thought. There was not a Divine Jesus who thought or felt differently than the Human Jesus. There is not human Jesus Christ who can be tempted separate from the Divine Jesus. There is only ONE Jesus Christ in that head of His, and it is a fully God-Man. We have to embrace this concept of the Union then. We can not simply attribute one aspect of Jesus to one nature, and another aspect to another nature, because Jesus Christ does not have multiple natures, multiple personalities, He is ONE.
So we have to say that the Divine Nature of Jesus Christ was equally tempted as was the humanity. Perhaps not tempted in the same kind of sense that we are tempted, that is to say, towards sin. However, whether we understand it or not, we need to accept that Jesus Christ, as God, was tempted by the Devil.
Here is my own speculation on the matter:
We read in the Old Testament of Moses or the Prophets apparently convincing, persuading, and even arguing with God. Clearly we have examples of the fully Divine God the Father expressing a similar ability as ourselves to be reasoned with. So if Satan is "temping" Jesus Christ, what is He tempting Jesus Christ to do? The Devil reminds Jesus of hunger, that is to say, the inherent weakness of humanity. Perhaps the Devil was asking Jesus to give up this Incarnation? The Devil tempts Jesus with lusts for power, saying that the death on the Cross would be unnecessary and that essentially the Devil might forefit the world to Jesus if Jesus foreits the Cross. Lastly, the Devil tempts Jesus to test the depth of His humanity by something similar to suicide in jumping off the ledge of pinnacle of the Temple. However, we know that Jesus Christ did not eat any earthly food without giving thanks in prayer, that He did not assert His authority or power but let it assert itself in Time, and that He did not even run to the Cross gleefully, but rather turned to the Father in the deep prayer in the Garden. The Devil asked Jesus to try to operate aside from the Will of the Father, to assert His own independence, to commit the same woeful error Satan committed before the Garden in his Fall. The Devil is not just tempting Jesus Christ as a human incarnation, rather I feel the Devil was trying to speak with the Godhead directly as Moses and the Prophets had done. Realistically I'd say its almost as if the Devil was trying to strike a bargain in his prayers to God (in the Person of Jesus Christ) just as many of us succumb to in our weakest moments in prayer.
I think it is too glib to say that Jesus Christ's Divinity was not tempted. Better to redefine what "temptation" is in the context of the Infinity of the Godhead, rather than to simply dismiss is as an impossibility. Surely God can change His own mind, just maybe that the bargain the Devil was trying to drive?