Question

If $(67^{67}+67)$ is divided by $68$, the remainder is: 

• A. 61
• B. 67
• C. 63
• D. 66

Hint:

When a base is “one less than the modulus” replace it by $-1$ (or $-k$) to simplify powers quickly.

Answer 1

The Correct Option is D

Step 1: Factor the expression as \[ 67^{67}+67=67\left(67^{66}+1\right). \]

Step 2: Since \( 67\equiv-1\pmod{68} \) and \( 66 \) is even, \( 67^{66}\equiv(-1)^{66}=1\pmod{68} \), so \( 67^{66}+1\equiv2\pmod{68} \).

Step 3: Multiplying back, \[ 67\left(67^{66}+1\right)\equiv67\times2=134\equiv134-68=66\pmod{68}. \]
\[ \boxed{66} \]