MathLabs

Problem 1

Let SS be a set of 9 distinct integers all of whose prime factors are at most 3. Prove that SS contains 3 distinct integers whose product is a perfect cube.
Step 6 of 6: Finish the proof
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ni1ni2ni3=a perfect cuben_{i_1}n_{i_2}n_{i_3}=\text{a perfect cube}
Detailed analysis

The three selected integers are distinct because they came from distinct elements of SS. Their type sum is (0,0)(0,0), so their product is a perfect cube, completing the proof.