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The average value of the smallest denominator of a rational number belongingto the interval $](j-1)/N,j/N]$, where~$j=1,\dots, N$, is proved to beasymptotically equivalent to~$16\pi^{-2}\sqrt{N}$, when $N$ tends to infinity.The result had been conjectured in 1977 by Kruyswijk and Meijer.

D{\'e}monstration d'une conjecture de Kruyswijk et Meijer sur le plus petit d{\'e}nominateur des nombres rationnels d'un intervalle