15. Solution: The side length of the wooden block is clearly the greatest common divisor of the length, width, and thickness of the wood. We use fractions of an inch as the unit of length, since
$$\begin{aligned} 357 & =3 \times 7 \times 17 \\ 105 & =3 \times 5 \times 7 \\ 84 & =2^{2} \times 3 \times 7 \end{aligned}$$

so $(357,105,84)=3 \times 7=21$.
Answer: The side length of the wooden block is 2 inches and 1 fraction of an inch.