Hello dear all staff of mathhelpforum and everyone,

I am a first time user and this is my first post threat.

I approached this question starting with the relation that

10 is congruent to (-1) (mod11), then with some calculation, it follows that:

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4311 is congruent to (-1)(mod 11) (1)

9! is congruent to 1(mod 11) (2)

16 is congruent to 5 (mod 11) (3)

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from (2) and (3), it follows that:

9!(16) is congruent to (1*5) (mod 11), (4)

combine (1) and (4) get,

(9!(16) +4311) is congruent to (5+(-1)) (mod 11)

= 4 (mod 11)

=> (9!(16) + 4311)^8603 is congruent to 4^8603 (mod 11).

the remainder i got was 4^8603.

This is obviously WRONG, but for now, I can't seem to recognize where the logic went wrong here.

Could someone help me point it out?

Thank you guys very much in advance