Question 549506
Several ways to do this. One way is to that


*[tex \LARGE 2^{1024} = 2(8^{341})], and that


*[tex \LARGE 2(8^{341}) \equiv 2((-1)^{341}) \equiv 2(-1) \equiv 7] (modulo 9).


Similarly,


*[tex \LARGE 5^{1024} = (5^4)(5^6)^{170} \equiv (5^4)(1^{170}) \equiv 5^4 \equiv 4] (modulo 9).


Hence the sum is equivalent to 7 + 4 + 1 = 12 = 3 (modulo 9), so the remainder is 3.