Question

What will be the remainder left when 2⁶⁸+3⁶⁸ is divided by 97?

• A. 0
• B. 96
• C. 42
• D. 1

Answer 1

The Correct Option is A

Remainder Calculation Using Fermat's Little Theorem 

We are asked to find the remainder when 2⁶⁸ + 3⁶⁸ is divided by 97.

- According to Fermat’s Little Theorem, if \( p \) is a prime number and \( a \) is an integer such that \( a \) and \( p \) are coprime, then:

\[ a^{p-1} \equiv 1 \pmod{p} \]

- Since 97 is a prime number, for \( a = 2 \) and \( a = 3 \), Fermat’s Little Theorem gives:

\[ 2^{96} \equiv 1 \pmod{97} \quad \text{and} \quad 3^{96} \equiv 1 \pmod{97} \]

- Since 68 < 96, we can calculate \( 2^{68} \) and \( 3^{68} \) modulo 97 using direct computation or by recognizing that they are valid under Fermat’s theorem conditions.

By direct calculation:

\[ 2^{68} + 3^{68} \equiv 0 \pmod{97} \]

Conclusion: The remainder when \( 2^{68} + 3^{68} \) is divided by 97 is 0.

The correct answer is (a) 0.