Question 148
Find the period of the repetend of the fraction $\frac{39}{1428}$ 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 finitely many steps, or it is of the form
$$
0.b_1b_2\cdots b_s a_1a_2\cdots a_k a_1a_2\cdots a_k\cdots
$$
Here the repeated sequence $a_1a_2\cdots a_k$ is called the *repetend* of the fraction, and the smallest length of the repetend, $k$, is called the *period* of the decimal number.)
(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finitely many steps, or it is of the form
$$
0.b_1b_2\cdots b_s a_1a_2\cdots a_k a_1a_2\cdots a_k\cdots
$$
Here the repeated sequence $a_1a_2\cdots a_k$ is called the *repetend* of the fraction, and the smallest length of the repetend, $k$, is called the *period* of the decimal number.)