Question

When \( 2^{56} \) is divided by 17, the remainder would be:

• A. 1
• B. 16
• C. 14
• D. None of these

Hint:

Use modular arithmetic and properties like Fermat’s Little Theorem to reduce large powers.

Answer 1

The Correct Option is C

We use modular arithmetic. Since \( 2^{56} \) is a large power, we reduce powers of 2 modulo 17. Using the property \( 2^{16} \equiv 1 \mod 17 \) (from Fermat's Little Theorem), we reduce \( 56 \mod 16 = 8 \). Thus: \[ 2^{56} \equiv 2^8 \mod 17. \] Now, calculate \( 2^8 \mod 17 \): \[ 2^8 = 256 \quad \text{and} \quad 256 \mod 17 = 14. \] Thus, the remainder is 14.