Let a_n = 6^n +8^n.
Determine the remainder on dividing a_83 by 49.
Thank you very much.
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?)