标题： 循环小数的长度 显示英文标题

标题： The length of the repeating decimal

摘要： 本文研究了分数表示为十进制形式时循环节的长度。 首先，详细解释了当分数的分母是质数的幂时，如何计算循环小数的长度。 然后，通过将分母分解为其质因数，并确定每个质因数的循环小数长度，得出总体循环小数长度是这些长度的最小公倍数。 此外，还研究了循环小数长度等于分母减1的条件，并讨论了这样的分数是否存在无限多个。 该主题与高斯在18世纪提出的一个未解问题有关，并且也与是否存在无限多个循环数的重要问题密切相关。 显示英文摘要

摘要： This paper investigates the length of the repeating decimal part when a fraction is expressed in decimal form. First, it provides a detailed explanation of how to calculate the length of the repeating decimal when the denominator of the fraction is a power of a prime number. Then, by factorizing the denominator into its prime factors and determining the repeating decimal length for each prime factor, the paper concludes that the overall repeating decimal length is the least common multiple of these lengths. Furthermore, it examines the conditions under which the repeating decimal length equals the denominator minus 1 and discusses whether such fractions exist in infinite quantity. This topic is connected to an unsolved problem posed by Gauss in the 18th century and is also closely related to the important question of whether cyclic numbers exist in infinite quantity.