Question

The remainder when \(27^{325}\) is divided by 14 is:

• A. \(11\)
• B. \(1\)
• C. \(10\)
• D. \(13\)

Hint:

Whenever a number is close to the divisor, replace it with its negative equivalent: \[ 13\equiv -1 \pmod{14}. \] This simplifies large powers immediately.

Answer 1

The Correct Option is D

Step 1: Reduce the base modulo 14. \[ 27\equiv 13 \pmod{14} \] Hence \[ 27^{325} \equiv 13^{325} \pmod{14} \] Since \[ 13\equiv -1 \pmod{14} \] we get \[ 13^{325} = (-1)^{325} \] \[ =-1 \]

Step 2: Convert the negative remainder into a positive remainder. \[ -1 \equiv 13 \pmod{14} \] Therefore, \[ \boxed{13} \]