GATE CSE 2019 | Question: 21

Question

The value of $3^{51} \text{ mod } 5$ is _____

Comments

@KUSHAGRA गुप्ता How did you took 51’s binary form and took it to 3’s power? can you explain why that works?

---

This is the same calculator btw. For this specific question we can get the answer. 

---

Perfect use of brain 🧠

Answer 1

3^51 mod 5

→ (3 mod 5)^51 mod 5

→ (-2)^51 mod 5

→ ( -2( -2 )^50) mod 5

→ (-2( 4 )^25) mod 5

→ (-2 ( 4 mod 5 )^25 ) mod 5

→ (-2 (-1)^25) mod 5

→ (-2*-1) mod 5

→ 2 mod 5

→ 2 ANSWER

Answer 2

(3)⁵¹ mod 5

3*(3)⁵⁰ mod 5

3*(3²)²⁵ mod 5

3*(9)²⁵ mod 5

3*(9 mod 5)²⁵ mod 5    [a mod n = a - kn mod n]

3*(-1)²⁵ mod 5

3*(-1) mod 5

-3 mod 5 = 2            [a mod n = a+n mod n]