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 4 of 6: Exclude a full row or column
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N(i,j)≤2;N(i,0)N(i,1)N(i,2)=0 or N(0,j)N(1,j)N(2,j)=0N(i,j)\le2;\quad N(i,0)N(i,1)N(i,2)=0\text{ or }N(0,j)N(1,j)N(2,j)=0
Detailed analysis

Assume no type occurs three times. If a row contains all three column types, choose one element from each: their first coordinates are equal and their second coordinates sum to 0+1+2=00+1+2=0 in F3\mathbb F_3, giving the required triple. The same applies to a full column. Thus no row or column may contain all three nonzero type boxes.