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no of distinct odd prime factors of n(n + 3) is a multiple of 3

Source: 2017 Grand Duchy of Lithuania, Mathematical Contest p4 (Baltic Way TST)

October 3, 2020
number theoryoddfactorsmultiple

Problem Statement

Show that there are infinitely many positive integers nn such that the number of distinct odd prime factors of n(n+3)n(n + 3) is a multiple of 33.
(For instance, 180=22⋅32⋅5180 = 2^2 \cdot 3^2 \cdot 5 has two distinct odd prime factors and 840=23⋅3⋅5⋅7840 = 2^3 \cdot 3 \cdot 5 \cdot 7 has three.)
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