Question

they died that we ..... live (may, might, could)

Answer 1

Hey!

The sentence will be :-

<> They died so that we might live.

As for the explanation, imagine there's a war going on. The soldiers are trying to protect you, but they died during the war, so that we might live. 'Might' offers the minimum possibility as the soldiers who died won't know that their sacrifice helped us or not.

Answer 2

They died that we might live.