when 2^2 is divided by 3 we have 1 as remainder
when 2^3 is divided by 3 we have 2 as remainder
when 2^4 is diviided by 3 we have 1 as remiander
when 2^5 is divided by 3 we have again 2 as remiander.
hence we can observer a pettern over here as when 2^even is diveded by 3 we get 1 remainder and when 2^odd is divided we get 2 as remainder.
in our question, we have 2^64 i.e. 2^even number hence, the remiander will be 1.