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Part of 2019 Turkey MO (2nd round)

Problems(1)

Number of divisors and an arithmetic progression

Source: Turkey National Mathematical Olympiad 2019 P2

12/23/2019
Let d(n)d(n) denote the number of divisors of a positive integer nn. If kk is a given odd number, prove that there exist an increasing arithmetic progression in positive integers (a1,a2,…a2019)(a_1,a_2,\ldots a_{2019}) such that gcd(k,d(a1)d(a2)…d(a2019))=1gcd(k,d(a_1)d(a_2)\ldots d(a_{2019})) =1
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