Question:
What is the value of $2^{100} \mod 5$?
What is the value of $2^{100} \mod 5$?
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When finding $a^n \mod m$, first determine the cycle length of powers of $a$ modulo $m$.
Updated On: Aug 1, 2025
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The Correct Option is A
Solution and Explanation
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
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