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For any positive integer 𝑛 , the Pseudo-Smarandache-Squarefree function Zw( 𝑛 ) is defined as the smallest positive integer m such that m 𝑛 is divisible by 𝑛 . That is, Zw( 𝑛 ) = min {m : m ∈ N, 𝑛 | m 𝑛 }. In reference [2], Felice Russo proposed many problems and conjectures related to the Pseudo-Smarandache-Squarefree function Zw( 𝑛 ) . The main purpose of this paper is using the elementary methods to study several problems in [2], and four of them are solved.