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Nearest Perfect Square Problem

Source: IOQM 2020 Question 25 (5 marks)

January 17, 2021
IOQMnumber theoryOlympiadOlympiads

Problem Statement

For a positive integer nn, let ⟨n⟩\langle n \rangle denote the perfect square integer closest to nn. For example, ⟨74⟩=81\langle 74 \rangle = 81, ⟨18⟩=16\langle 18 \rangle = 16. If NN is the smallest positive integer such that ⟨91⟩⋅⟨120⟩⋅⟨143⟩⋅⟨180⟩⋅⟨N⟩=91⋅120⋅143⋅180⋅N \langle 91 \rangle \cdot \langle 120 \rangle \cdot \langle 143 \rangle \cdot \langle 180 \rangle \cdot \langle N \rangle = 91 \cdot 120 \cdot 143 \cdot 180 \cdot N find the sum of the squares of the digits of NN.
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