Question

Find the remainder when 2⁸¹ is divided by 17.

Answer 1

2⁸¹ ≡ x (mod 17)

2⁴⁰ × 2⁴⁰ × 2⁴¹ ≡ x (mod 17)

(2⁴)¹⁰ × (2⁴)¹⁰ × 2¹ ≡ x (mod 17)

(16)¹⁰ × (16)¹⁰ × 2 ≡ x (mod 17)

(16⁵)² × (16⁵)² × 2

(16⁵) ≡ 16 (mod 17)

(16⁵)² ≡ 16² (mod 17)

(16⁵)² ≡ 256 (mod 17)

= 1 (mod 17)                  ...[∵ 255 is divisible by 17]

(16⁵)² × (16⁵)² × 2 ≡ 1 × 1 × 2 (mod 17)

∴ 2⁸¹ ≡ 2 (mod 17)

∴ x = 2