Find the remainder when $10^{400}$ is divided by 199? I am trying to solve a problem 

Find the remainder when the $10^{400}$ is divided  by 199?

I tried it by breaking $10^{400}$ to $1000^{133}*10$ .

And when 1000 is divided by 199 remainder is 5.

So finally we have to find a remainder of :

$5^{133}*10$ 

But from here I could not find anything so that it can be reduced to smaller numbers.

How can I achieve this?

Is there is any special defined way to solve this type of problem where denominator is a big prime number?

Thanks in advance.
You can use Fermat's little theorem. It states that if $n$ is prime then $a^n$ has the same remainder as $a$ when divided by $n$.

So, $10^{400} = 10^2 (10^{199})^2$. Since $10^{199}$ has remainder $10$ when divided by $199$, the remainder is therefore the same as the remainder of $10^4$ when divided by $199$. $10^4 = 10000 = 50*199 + 50$, so the remainder is $50$.
Since 199 is prime and gcd(10,199) = 1

So, $10^{198} \equiv 1 (mod 199)$

Squaring the both side: $10^{396} \equiv 1 (mod 199)$

Now: $10^3 \equiv 5(mod199)$ 

$10^{4} \equiv 50 (mod 199)$

$10^{400} \equiv 50 (mod 199)$

So, the remainder is 50. This method is known as Euler's Totient