The greatest common divisor of two integers is $(x+2)$ and their least common multiple is $x(x+2)$, where $x$ is a positive integer. If one of the integers is 24, what is the smallest possible value of the other one?
We know that $\gcd(m,n) \cdot \mathop{\text{lcm}}[m,n] = mn$ for all positive integers $m$ and $n$.  Hence, in this case, the other number is \[\frac{(x + 2) \cdot x(x + 2)}{24} = \frac{x(x + 2)^2}{24}.\]To minimize this number, we minimize $x$.

This expression is not an integer for $x =$ 1, 2, or 3, but when $x = 4$, this expression is $4 \cdot 6^2/24 = 6$.

Note that that the greatest common divisor of 6 and 24 is 6, and $x + 2 = 4 + 2 = 6$.  The least common multiple is 24, and $x(x + 2) = 4 \cdot (4 + 2) = 24$, so $x = 4$ is a possible value.  Therefore, the smallest possible value for the other number is $\boxed{6}$.