Question

When $5^{20}$ is divided by 7, the remainder is

• A. $1$
• B. $3$
• C. $4$
• D. $6$

Hint:

Use Fermat's little theorem or modular exponentiation.

Answer 1

The Correct Option is A

Step 1: $5 \equiv -2 \pmod{7}$, so $5^{20} \equiv (-2)^{20} = 2^{20} \pmod{7}$.}
Step 2: $2^3 = 8 \equiv 1 \pmod{7}$, so $2^{20} = 2^{3\cdot6+2} = (2^3)^6 \cdot 2^2 \equiv 1 \cdot 4 \equiv 4 \pmod{7}$.}
Step 3: Remainder is 4. Option (C).}