An easy one!

$\Large F(x)=x^{13}+1$ Then the remainder when F(x) is divided by x-1 is F(1)..( $\color{darkviolet}{\text{Remainder Theorem}}$ ) $\Large F(1)=2$ Hence the remainder when $x^{13}+1$ is divided by x-1 is $\large\boxed{2}$

$x^{13}$ +1 = $(x-1+1)^{13}$ +1

Using BINOMIAL EXPANSION

The only term in $(x-1+1)^{13}$ not divisible by (x-1) is ${13\choose0}$ ×1.

So when $(x-1+1)^{13}$ +1 is divided by (x-1) the remainder is 1+1 = $\boxed{2}$