jmc

Using long division, we find that $10,000$ divided by 15 yields a quotient of 666 with a remainder of 10. Thus $10,005$ is a multiple of 15, and $\boxed{10,010}$ is the least five-digit integer which is congruent to 5 (mod 15). For confirmation, note that $10,010-15=9,995$ is the next-highest integer which is congruent to 5 (mod 15).