Question

Let $N=3^n+8^n+{10}^n$. When $n$ is even, the remainder when $N$ is divided by $9$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (C)

$2$

Any power of $3$ more than $1$ is divisible by $9$, so corresponding remainder is $0$.
When $8$ is divided by $9$, the remainder is $-1$.
Thus, for an even power, the corresponding remainder is ${(-1)}^{even}=1$.
When $10$ is divided by $9$, the remainder is $1$.
Thus, for an even power, the corresponding remainder is ${\left(1\right)}^{even}=1$.
Hence,total remainder $=0+1+1=2$
Hence, (C) is correct.