a_83=6^83+8^83

.......=(7-1)^83 + (7+1)^83

Now expand the powers using the binomial theorem:

a_83=[(-1)^83 + 83 (-1)^82 7 + K_1 49] + [(1)^83 + 83 (1)^82 7 + K_2 49]

for some integers K_1 and K_2. So for some integer K_3

a_83=2 83 7 + K_3 49

.......= 1162 + K_3 49 = (23 49 + 35) + K_3 49

So the required remainder is 35.

RonL

(What does AIME stand for?)