Question

Let $N=3^n+8^n+{10}^n$,Given n is odd, the remainder when N is divided by $13$ is

• A. $2$
• B. $5$
• C. $8$
• D. None of these

Answer 1

Answer: (B), (C)

The correct options are
B $5$
C $8$

We know that:
$3^n+{10}^n$ is divisible by $13$ for odd value of $n$.
Again:
$8^2$ when divided by $13$ leaves remainder $-1$
Thus, if $n=4m$ then the remainder $={(-1)}^{2m}=1$
So, if $n=4m+1$ then the remainder $=1\ast 8=8$
And, if $n=4m+3$ then the remainder $=1\ast 8^3=>5$
Thus, remainder could be $5$ or $8$ when n is odd.
Hence, (B) and (C) are correct.