Solution: 2022-21 A determinant of greatest common divisors

Let \(\varphi(x)\) be the Euler’s totient function. Let \(S = \{a_1,\dots, a_n\}\) be a set of positive integers such that for any \(a_i\), all of its positive divisors are also in \(S\). Let \(A\) be the matrix with entries \(A_{i,j} = gcd(a_i,a_j)\) being the greatest common divisors of \(a_i\) and \(a_j\). Prove that \(\det(A) = \prod_{i=1}^{n} \varphi(a_i)\).

The best solution was submitted by Noitnetta Yobepyh (Snaejwen High School, +4). Congratulations!

Here is the best solution of problem 2022-21.

Other solutions were submitted by 기영인 (KAIST 22학번, +3), 여인영 (KAIST 물리학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정, +3), 전해구 (KAIST 기계공학과 졸업생, +2), 최예준 (서울과기대 행정학과 21학번, +2).

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