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We establish estimates for the number of ways to represent any reducedresidue class as a product of a prime and an integer free of small primefactors. Our best results is conditional on the Generalised Riemann hypothesis(GRH). As a corollary, we make progress on a conjecture of Erd\"os, Odlyzko andS\'ark\"ozy.

Product of an integer free of small prime factors and prime in arithmetic progression