Question

What is the value of $2^{100} \mod 5$? 
 

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

When finding $a^n \mod m$, first determine the cycle length of powers of $a$ modulo $m$.

Answer 1

The Correct Option is A

Step 1: Find the Pattern in Powers of 2 Modulo 5

Let's compute the first few powers of 2 modulo 5:

21 mod 5 = 2  
22 mod 5 = 4  
23 mod 5 = 3  
24 mod 5 = 1
    

We observe that the remainders repeat every 4 powers:

Cycle: 2, 4, 3, 1, 2, 4, 3, 1, ...

Step 2: Use the Cycle Length

Since the cycle length is 4, we find:

100 mod 4 = 0
    

This means 2¹⁰⁰ corresponds to the 4th term in the cycle.

Step 3: Find the Remainder

From the cycle (2, 4, 3, 1), the 4th term is 1.

Final Answer:

2¹⁰⁰ mod 5 = 1