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Kapron, Khodakhast Bibak, L\\'aszl\\'o T\\'oth, Roberto Tauraso, Venkatesh Srinivasan","submitted_at":"2015-03-05T22:23:32Z","abstract_excerpt":"In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence $a_1x_1+\\cdots +a_kx_k\\equiv b \\pmod{n}$, with $\\gcd(x_i,n)=t_i$ ($1\\leq i\\leq k$), where $a_1,t_1,\\ldots,a_k,t_k, b,n$ ($n\\geq 1$) are arbitrary integers. As a consequence, we derive necessary and sufficient conditions under which the above restricted linear congruence has no solutions. 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