The positive integer divisors of 24 are $1,2,3,4,6,8,12,24$. The pairs of divisors that give a product of 24 are $24 \times 1,12 \times 2,8 \times 3$, and $6 \times 4$. We want to find two positive integers $x$ and $y$ whose product is 24 and whose difference is 5. Since $8 \times 3=24$ and $8-3=5$, then $x=8$ and $y=3$ are the required integers. Here, $x+y=8+3=11$.