Question

Find the remainder when $354^{176 - 346^{176}$ is divided by 700.

• A. \(5 \)
• B. \(0 \)
• C. \(2 \)
• D. \(4 \)
• E. \(3 \)

Hint:

If $a \equiv -b \pmod{m}$ and power is even: \[ a^n \equiv b^n \pmod{m} \] This helps simplify large exponents quickly.

Answer 1

The Correct Option is B

Concept: Use modular arithmetic and factorization: \[ a^n - b^n = (a-b)(\cdots) \] If $(a-b)$ is divisible by the divisor, then the whole expression may be divisible.
Step 1: Find difference.
\[ 354 - 346 = 8 \]

Step 2: Check divisibility by 700.
\[ 700 = 7 \times 100 = 7 \times 4 \times 25 = 2^2 \times 5^2 \times 7 \] Now observe: \[ 354 \equiv -346 \pmod{700} \]

Step 3: Use symmetry.
\[ 354^{176} - 346^{176} \equiv (-346)^{176} - 346^{176} \pmod{700} \] Since 176 is even: \[ (-346)^{176} = 346^{176} \] \[ \Rightarrow 346^{176} - 346^{176} = 0 \]

Step 4: Final result.
\[ \boxed{0} \]