Question

What is the remainder when $7^{100}$ is divided by 8? 
 

• A. 1
• B. 3
• C. 5
• D. 7

Hint:

When a number is 1 less than the modulus, use $a \equiv -1$ (mod m) to simplify powers. Even exponents give remainder 1, odd give modulus-1.

Answer 1

The Correct Option is A

- Step 1: Understanding the problem - We are asked to find the remainder when $7^{100}$ is divided by 8. 

This is a modular arithmetic problem. 

- Step 2: Observing the base relative to the modulus - Note that $7 \equiv -1 \ (\text{mod}\ 8)$, 

since $7$ is exactly one less than $8$. 

- Step 3: Simplifying using this congruence - \[ 7^{100} \equiv (-1)^{100} \ (\text{mod}\ 8) \] 

Since the exponent $100$ is even, $(-1)^{100} = 1$. 

Therefore: \[ 7^{100} \equiv 1 \ (\text{mod}\ 8) \] 

- Step 4: Alternate check via pattern - Calculate small powers: $7^1 \equiv 7$, $7^2 \equiv 1$, $7^3 \equiv 7$, $7^4 \equiv 1 \ (\text{mod}\ 8)$. 

Pattern repeats every 2 powers. Since $100$ is even, remainder = $1$. 

- Step 5: Conclusion - The remainder is 1, so the correct answer is option (1).