Question

The total number of two digit numbers $n$, such that $3^n+7^n$ is a multiple of $10$, is

Answer 1

$\because 7^n=(10-3{)}^n=10k+(-3{)}^n$
$\therefore f=7^n+3^n=10k+(-3{)}^n+3^n$
So, if $n$ is odd, $f=10k$
if $n$ is even, $f=10k+2\cdot 3^n$
Let $n=2t;t\in N$
$3^n=3^{2t}=(10-1{)}^t$
$=10p+(-1{)}^t$
$=10p\pm 1$
$\therefore$ if n = even then $7^n+3^n$ wil not be multiple of 10
So if n is odd then only $7^n+3^n$ will be multiple of 10
$\Rightarrow n=11,13,15,............,99$
$\therefore n=45$