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Let m and n be positive integers. Let (mn)=m!n!(m−n)! denote the binomial coefficient indexed by m and n, where n! is the factorial of n. For any prime p, let νp(n) denote the largest nonnegative integer r such that pr divides n. In this paper, we use the p-adic method to show the following identity:gcd({(mnk):1≤k≤mn,gcd(k,m)=1})=m∏primep|gcd(m,n)pνp(n).This extends greatly the identities obtained...
Let f be an arithmetic function and S = {x1, …, xn} be a set of n distinct positive integers. By (f(xi, xj)) (resp. (f[xi, xj])) we denote the n × n matrix having f evaluated at the greatest common divisor (xi, xj) (resp. the least common multiple [xi, xj]) of x, and xj as its (i, j)-entry, respectively. The set S is said to be gcd closed if (xi, xj) ∈ S for 1 ≤ i, j ≤ n. In this paper, we give formulas...
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