$\text { Use the fact that } 29 = 16 + 8 + 4 + 1 \text { to compute } 18 ^ { 29 } \bmod 65 \text {. }$

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The Answer is

18 ^ { 1 } \bmod 65 = 18

18 ^ { 2 } \bmod 65 = 324 \bmod 65 = 64 \bmod 65

18 ^ { 4 } \bmod 65 = 64 ^ { 2 } \bmod 65 = 4096 \bmod 65 = 1

18 ^ { 8 } \bmod 65 = 1 ^ { 2 } \bmod 65 = 1

18 ^ { 16 } \bmod 65 = 1 ^ { 2 } \bmod 65 = 1

Hence, by Theorem 8.4.3,

18 ^ { 29 } = 18 ^ { 16 + 8 + 4 + 1 } = 18 ^ { 16 } 18 ^ { 8 } 18 ^ { 4 } 18 ^ { 1 } \equiv 1 \cdot 1 \cdot 1 \cdot 18 \equiv 18 ( \bmod 65 )

ext { and thus } 18 ^ { 29 } \bmod 65 = 18

Essay