Question

When ${67}^{67}+67$ is divided by $68$, the remainder is:

• A. $0$
• B. $22$
• C. $33$
• D. $66$

Answer 1

Answer: (D)

$66$

$\Rightarrow$ $(x^n+1)$ is divisible by $(x+1)$ only when $n$ is odd.

$\Rightarrow$ $({67}^{67}+1)$ is divisible by $(67+1)$

$\Rightarrow$ $({67}^{67}+1)$ is divisible by $68$.

$\Rightarrow$ $({67}^{67}+1)÷68$ gives a remainder of $0$.

$\Rightarrow$ $\left[\right({67}^{67}+1)+66]÷68$ gives a remainder of $66$.

$\therefore$ $({67}^{67}+67)÷68$ gives a remainder of $66$.