How many positive multiples of 13 are three-digit integers?
We're looking for the number of positive multiples of 13 between 99 and 1000. $13 \times 7 = 91 < 100 < 104 = 13 \times 8$ and $13 \times 76 = 988 < 1000 < 1001 = 13 \times 77$, so the three-digit multiples of 13 are $$13\times8,13\times9,\ldots,13\times75,13\times76.$$ The number of terms in this list is the same as the number of terms in the list $$8,9,\ldots,75,76.$$ Subtracting $7$ from every term in the previous list, we get $$1,2,\ldots,68,69.$$ Therefore, there are $\boxed{69}$ positive multiples of 13 that are three-digit integers.