If $13^{50}$ is divided by $14$, the remainder is

Using modular arithmetic concepts:
$$13\equiv -1(\mathrm{mod}14)$$
Raise both sides to the power of $50$:
$$13^{50}\equiv (-1{)}^{50}(\mathrm{mod}14)$$
Since an even power of $-1$ is $+1$:
$$13^{50}\equiv 1(\mathrm{mod}14)$$
Hence, the remainder when $13^{50}$ is divided by $14$ is $1$.