Question

Find the remainder obtained when $5^{2009}+{13}^{2009}$ is divided by $18$.

• A. $1$
• B. $8$
• C. $0$
• D. $18$

Answer 1

Answer: (C)

$0$

Let,
$f\left(x\right)=x^n+y^n$ and
$g\left(x\right)=x+y$
$=>x+y=0$
$=>x=-y$
If $f\left(x\right)$ is divided by $g\left(x\right)$ then,$f(-y)$ should be zero.
Therefore,
$f(-y)=(-y{)}^n+y^n$
$=0$ if $n$ is odd
$\ne 0$ if $n$ is even
Here,
$x=5$
$y=13$
$n=2009$ and
$x+y=18$ , $n$ is odd.
Thus,
$5^{2009}+{13}^{2009}$ is divisible by $18$ and remainder is $0$