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product (a_1 - 1)(a_2 - 2) ... (a_n -n) is an even number

Source: Eotvos 1906 p3

September 6, 2024
number theoryEvenpermutation

Problem Statement

Let a1,a2,...,ana_1, a_2, ...,a_n represent an arbitrary arrangement of the numbers 1,2,...,n1, 2, ...,n. Prove that, if nn is odd, the product (a1−1)(a2−2)...(an−n)(a_1 - 1)(a_2 - 2) ... (a_n -n) is an even number.
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