Let a_n = 6^n +8^n.

Determine the remainder on dividing a_83 by 49.

Thank you very much.

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- December 25th 2006, 10:16 PMJenny201983 AIME question
Let a_n = 6^n +8^n.

Determine the remainder on dividing a_83 by 49.

Thank you very much. - December 25th 2006, 11:46 PMCaptainBlack
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?) - December 26th 2006, 01:47 AMJenny20
Hi Captainblack,

Could you please solve this question by using the modulo 49? Thank you very much.

for example:

6^83 + 8^83 is congruent to x ( mod 49)

I don't know how to carry on the next working step after this.... - December 26th 2006, 02:01 AMCaptainBlack
- December 26th 2006, 02:10 AMCaptainBlack
- December 26th 2006, 02:22 AMJenny20
a_83=2 83 7 mod 49

.......= 1162 mod 49

=======================

Hi Captainblack,

I don't get the above step = > 2837 mod 49 equal to 1162 mod 49.

In fact, I have completely forgotten the binomial thoerem. So feel confused with that part of your work. :( - December 26th 2006, 02:24 AMCaptainBlack
- December 26th 2006, 02:37 AMJenny20
ic Thank you very much. :)