To get the unit digit of $7^{78}$, we will find $7^{78}\;mod\;10.$
$7^{78}\; mod\;10=7^{76+2}\;mod\;10$
$=(7^{76}*7^{2})\;mod\;10$
$=(7^{76}\;mod\;10*7^{2}\;mod\;10)\;mod\;10$
$=((7^{4})^{19}\;mod\;10*7^{2}\;mod\;10)\;mod\;10$
$=(1*7^{2}\;mod\;10)\;mod\;10$ ( $\because$ when $a,n$ are co-primes then $a^{\Phi (n)}mod\;n=1$. Here $7,10$ are co-primes and $\Phi (10)=4$. So, $7^{4}\;mod\;10=1$ ) $[\Phi(n) = Euler \;\;Totient\;\; Function ]$
$=(49\;mod\;10)\;mod 10$
$=9\;mod\;10$
$= 9$