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D. Hensley showed in 1994 that the number of steps taken by the Euclideanalgorithm to find the greatest common divisor of two natural numbers less thanor equal to n follows a normal distribution in the limit as n tends toinfinity. V. Baladi and B. Vall\'ee subsequently gave an alternative proof forboth the classical Euclidean algorithm and several of its close variants, basedon a detailed investigation of spectral properties of the transfer operatorassociated to the Gauss map, building on deep results of D. Dolgopyat. In thisarticle we give a much shorter, albeit less quantitative, proof of this resultusing only basic spectral properties of the transfer operator together with themethod of moments and a Tauberian theorem due to H. Delange.

A short proof that the number of division steps in the Euclidean algorithm is normally distributed