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𝑛=∏{𝑝}𝑝^{𝑒{𝑝}}

2𝑛³=𝑘² ⟹ ∀𝑝, 𝑒_{𝑝}(2𝑛³) ≡0 (mod 2)

3𝑛²=𝑚³ ⟹ ∀𝑝, 𝑒_{𝑝}(3𝑛²) ≡0 (mod 3)

𝑝=2: 1+3𝑎 ≡0 (mod 2) ⟹ 𝑎 ≡1 (mod 2)

𝑝=3: 3𝑏 ≡0 (mod 2) ⟹ 𝑏 ≡0 (mod 2)

𝑝≠2,3: 3𝑐 ≡0 (mod 2) ⟹ 𝑐 ≡0 (mod 2)

𝑝=2: 2𝑎 ≡0 (mod 3) ⟹ 𝑎 ≡0 (mod 3)

𝑝=3: 1+2𝑏 ≡0 (mod 3) ⟹ 𝑏 ≡1 (mod 3)

𝑝≠2,3: 2𝑐 ≡0 (mod 3) ⟹ 𝑐 ≡0 (mod 3/ gcd(2,3)) , 𝑐=2𝑑 ⟹ 4𝑑 ≡0 (mod 3) ⟹ 𝑑 ≡0 (mod 3) ⟹ 𝑐 ≡0 (mod 6)

min 𝑎: 𝑎≡1 (mod 2), 𝑎≡0 (mod 3) ⟹ 𝑎=3

min 𝑏: 𝑏≡0 (mod 2), 𝑏≡1 (mod 3) ⟹ 𝑏=4

∀𝑝≠2,3: min 𝑐=0 or 6, but 𝑝^6 >1 ⟹ larger 𝑛

⟹ min 𝑛=2³⋅3⁴=8⋅81=648