Question

4. What are the last TWO DIGITS of 2^2017

Answer 1

Compute 2²⁰¹⁷ modulo 100.

From the given table,

2²² ≡ 2⁴² ≡ 2⁶² ≡ 4 (mod 100)

and we have

2017 = 32 • 62 + 33

62 = 1 • 42 + 20

Then

2²⁰¹⁷ ≡ (2⁶²)³² • 2³³ (mod 100)

2²⁰¹⁷ ≡ 4³² • 2³³ (mod 100)

2²⁰¹⁷ ≡ (2²)³² • 2³³ (mod 100)

2²⁰¹⁷ ≡ 2⁶⁴ • 2³³ (mod 100)

2²⁰¹⁷ ≡ 2⁶² • 2²² • 2¹³ (mod 100)

2²⁰¹⁷ ≡ 4² • 2¹³ (mod 100)

2²⁰¹⁷ ≡ (2²)² • 2¹³ (mod 100)

2²⁰¹⁷ ≡ 2⁴ • 2¹³ (mod 100)

2²⁰¹⁷ ≡ 2¹⁷ (mod 100)

and from the table, we get

2²⁰¹⁷ ≡ 2¹⁷ ≡ 72 (mod 100)