Modular Arithmetic:
$a \equiv b(mod n)$ (i.e,. a and b leaves same remainder when divided by n), then
$a^k \equiv b^k(mod n)$, where k be any non-negative integer
Division Algorithm:
$dividend = divisor * quotient + remainder$
Let's find out the number which is modulo congruent to 13
$13 = 17 * 1 + (-4)$
Therefore, $13 \equiv (-4) mod 17$
which implies $13^{99} \equiv (-4)^{99} mod 17$
$13^{99} \equiv (-4)*((-4)^2)^{49} mod 17$
Value of $16 mod 17 = -1$ which means $16 \equiv -1(modn)$ so that we can replace 16 with -1 in the above which makes our calculation easy
$13^{99} \equiv (-4)*(-1)^{49} mod 17$
$13^{99} \equiv (-4)*(-1)mod 17$
$13^{99} \equiv 4 mod 17$
Therefore, the value of $13^{99} mod 17$ is 4
References:
Modular Arithmetic Wikipedia