The total number of two digit numbers ' n ' , such that 3 n + 7 n is a multiple of 10 , is ___ .

$\because 7^n=(10-3{)}^n=10k+(-3{)}^n$

$7^n+3^n=10k+(-3{)}^n+3^n$

$10k+{\left(-3\right)}^n+{\left(3\right)}^n=\left\{\begin{array}{l}10k \text{if n=odd}\\ 10k+2.3^n \text{if n=even}\end{array}\right.$

Let $n=2t; t\in N$

$\therefore 3^n=3^{2t}=(10-1{)}^t$

$=10p+(-1{)}^t$

.$=10p\pm 1$

$\therefore$ if $n=$ even then $7^n+3^n$ will not be multiply of $10$

So if $n$ is odd then only $7^n+3^n$ will be multiply of $10$

$\therefore n=11,13,15,\dots \dots \dots .,99$

$\therefore$ Number of odd two digit numbers $=45$