A box has the shape of a rectangular block with dimensions $102 \mathrm{~cm}, 255 \mathrm{~cm}$, and $170 \mathrm{~cm}$. We want to store the smallest possible number of small cubes with integer edge lengths in this box, so as to fill the entire box.

Hint: Note that the length of the edge of the cube must be a divisor of each of the three dimensions of the box.

(a) What is the length of the edge of each block?

(b) How many blocks will be needed?

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(a) Since the number of blocks is the smallest possible, the edge of the same must be the largest possible. The measure of the edge must be a divisor of 102, 255, and 170. Since we want the largest possible edge, the measure of it must be equal to $\operatorname{gcd}(102,255,170)=17$. Therefore, the edge of the cube measures $17 \mathrm{~cm}$.

(b) The number of blocks is

$$
\frac{102 \cdot 255 \cdot 170}{17 \cdot 17 \cdot 17}=6 \cdot 15 \cdot 10=900
$$

Suggestion: Determine the minimum value for the sum of the digits of the number.

Facts that Help: The sum of the digits of a multiple of 9 is divisible by 9.

Suggestion: Determine the possible values for the product and their factorizations.

Facts that Help: 101 is prime.
Suggestion: Initially show that he could not have bought more than 127 items.