A maths question (difficult)

(for positive integer number n)
1. lim(n->∞) (133^n+110^n+84^n+27^n)/144^n =0
so, n is not a large number.
in fact, if n>6, 144^n > 133^n+110^n+84^n +27^n
2. consider the unit digit of 133^n, 110^n, 84^n, 27^n, 144^n
then n must be 1,3, 5
case n=1: reject.
case n=3: reject.
csee n=5: match
Ans: n=5


2010-02-22 01:52:05 補充：
133^n+110^n+84^n+27^n≡ 144^n (mod 7)
0+2^n+0+(-1)^n ≡ 4^n (mod 7)
when n=1, 1≡4 (mod 7) (inconsistent)
when n=3, 8-1≡ 64 (mod 7) (inconsistent)
when n=7, 2-1≡ 4^7≡4 (mod 7) (inconsistent)
when n>7, 144^n > 133^n+110^n+84^n+27^n
so, n=5

there is no algebraic method
you have to use numerical method