Find the smallest number by which 1000 must be divided to make the quotient a perfect cube. (1000 is already a perfect cube.)

For non-perfect cubes, bracket the cube root between two consecutive integers.

Example: ∛100. We know $4^3=64$ and $5^3=125$. So 4 < ∛100 < 5. Closer to 5 (100 is closer to 125 than 64). Refined estimate: ∛100 ≈ 4.64.

For perfect cubes, look at the last digit (last digit rule for cubes: each digit cubes to a unique last digit). 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9, 0→0.