Prime factor of a composite three digit integer

Either the question as given to you is erroneous, or you copied it incorrectly. I will assume this is the former rather than the latter.

Remember that 2 is a prime number, too. So if $n = 2p$, with $p$ a prime number, then, for sufficiently large $n > 4$, we would see 2 is much smaller than $\sqrt n$, and $p$ is much greater.

The $n$ we're looking for is much greater than 99 but certainly less than 1000. We see that $999 = 3^3 \times 37$, but $998 = 2 \times 499$. We have $\sqrt{998} \approx 31.5911$, and the least prime factor, 2, is indeed less than 31. But 499 is certainly greater than 32.

So you're right: as you have the question, the correct answer is indeed 499.

Just to be absolutely sure: could the answer be 503? No, because $2 \times 503 = 1006$, which has four digits.

If the question is:

What is the largest possible minimum prime factor of a composite three digit integer?

(or something similar) then the answer as given by the book, $31$, is correct, because any composite number must have a factor less than or equal to its square root.

Otherwise, as already observed, a number of the form $2p$ with $p$ prime will give you the largest prime factor of a composite, in this case $499$.