jmc

The first ten positive multiples of 13 are 13, $13\cdot 2$, $\dots$, $13\cdot 10$, so we want to find the sum $13+13\cdot 2+\cdots +13\cdot 10=13\cdot (1+2+\cdots +10)$.

For all $n$, $1+2+\cdots +n=n(n+1)/2$, so $13\cdot (1+2+\cdots +10)=13\cdot 10\cdot 11/2=\boxed{715}$.