Problem 1

Let

S

be a set of 9 distinct integers all of whose prime factors are at most 3. Prove that

S

contains 3 distinct integers whose product is a perfect cube.

Step 1 of 6: Assign each integer a type

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n_i=\varepsilon_i2^{a_i}3^{b_i},\qquad \varepsilon_i\in\{-1,1\},\qquad t(n_i)=(a_i\bmod3,b_i\bmod3)\in\mathbb F_3^2

Detailed analysis

Write every element as $n_i=\varepsilon_i2^{a_i}3^{b_i}$ with $\varepsilon_i\in\{-1,1\}$ and nonnegative exponents. Its type is $t(n_i)=(a_i\bmod3,b_i\bmod3)$ , one of the nine points of $\mathbb F_3^2$ ; the sign does not affect whether a product is a perfect cube.