How to determine the remainder of $ \frac{7^{369}}{350} $? How to determine the remainder of the following :
$$ \frac{7^{369}}{350} $$
Is there any tricky way to determine this?
I start as $$\frac{7^{368}}{50} \ \ , $$ 
This type of problem are highly asked in aptitude exam.

One thing you can sometimes do with problems like this is to find the remainder modulo the different prime powers and then piece the information together to find the original remainder. Essentially this amounts to using the Chinese Remainder Theorem which you may not have learned yet but one doesn't really need to know it here. 
Since $350 = 2\cdot 5^2 \cdot 7$ we compute
$$ 7^{369} \equiv 1^{369}=1 \pmod{2} $$
and 
$$ 7^{369} = 7\cdot (7^2)^{184} \equiv 7\cdot(-1)^{184} = 7\cdot 1 = 7\pmod{5^2}$$
and
$$ 7^{369} \equiv 0^{369}=0\pmod{7}.$$
Now only possible remainders $r< 350$ sastisfying $r\equiv 7 \pmod{5^2}$ are $7, 25+7, \ldots, 325+7$. Since we must also have $r\equiv 0 \pmod{7}$ we are only left with $r=7$ or $r=7\cdot 25+7=182$. Finally using $r\equiv 1\pmod{2}$, we see that $r=7$.  
