• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.863-872
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• 2023

Let n ⩾ 2 be an integer, we denote the smallest integer b such that gcd {(ⁿ_k) : b < k < n - b} > 1 as b(n). For any prime p, we denote the highest exponent α such that p^α | n as v_p(n). In this paper, we partially answer a question asked by Hong in 2016. For a composite number n and a prime number p with p | n, let n = a_mp^m + r, 0 ⩽ r < p^m, 0 < a_m < p. Then we have $$v_p\(\text{gcd}\{\(n\\k\)\;:\;b(n)1\}\)=\{\array{1,&&a_m=1\text{ and }r=b(n),\\0,&&\text{otherwise.}}$$