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

Work modulo $68$. Since $67\equiv -1\pmod{68}$ and the exponent is odd, \[ 67^{67}\equiv (-1)^{67}\equiv -1\pmod{68}. \] Therefore, \[ 67^{67}+67\equiv (-1)+(-1)\equiv -2\equiv 68-2=\boxed{66}\pmod{68}. \]