Problem

Source: 2023 Hong Kong TST 3 (CHKMO)

Tags: number theory



Find the period of the repetend of the fraction 391428 by using binary numbers, i.e. its binary decimal representation. (Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form 0.b1b2bsa1a2aka1a2aka1a2ak. Here the repeated sequence a1a2ak is called the repetend of the fraction, and the smallest length of the repetend, k, is called the period of the decimal number.)