To solve the problem of finding the number of distinct ways to paint a cube with 3 colors (considering rotations as identical), we use Burnside's Lemma, which states the number of distinct colorings equals the average number of colorings fixed by each rotational symmetry of the cube.

Step 1: List Rotational Symmetries of Cube

There are 24 total rotational symmetries, grouped into 5 types:

1. Identity Rotation (1 element)

Fixes all colorings: $3^6 = 729$.

2. 90°/270° face-axis rotations (6 elements)

Each rotation cycles 4 lateral faces, so fixed colorings: $3^3 = 27$ (top, bottom, and 4 lateral faces same). Total contribution: $6×27 = 162$.

3. 180° face-axis rotations (3 elements)

Splits lateral faces into 2 pairs, fixed colorings: $3^4 = 81$. Total contribution: $3×81 = 243$.

4. 180° edge-axis rotations (6 elements)

Splits faces into 3 pairs, fixed colorings: $3^3 = 27$. Total contribution: $6×27 = 162$.

5. 120°/240° vertex-axis rotations (8 elements)

Cycles 3 faces each, fixed colorings: $3^2 =9$. Total contribution: $8×9 =72$.

Step 2: Calculate Average Fixed Colorings

Sum of fixed colorings: $729 +162 +243 +162 +72 =1368$.
Divide by 24: $\frac{1368}{24}=57$.

Answer: $\boxed{57}$

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