Why **Excursion length** is less than **Denominator**?

Question

Can anyone prove that the Excursion length of any fraction's decimal representation which is cyclic, is less than its Denominator?

Answer 1

Let us consider $\frac17$ as an example. Successive digits of this fraction are determined by division with remainder. Since 7 does not fit into 1, we start with $$ 0. $$ with a remainder of 1. Then for the second digit, we multiply this remainder by 10, giving 10, and repeating: 7 fits into 10 once, with remainder 3. We obtain $$ 0.1 $$ and we repeat. 7 fits into 30 four times, with remainder 2; into 20 twice, with remainder 6; into 60 eight times, with remainder 4; into 40 five times, with remainder 5; and into 50 seven times with remainder 1. This gives $$ 0.142857 $$ and now we are back to the start of remainder 1. Thus the number starts repeating. The repeating part could never have been longer than 7, because there are only 7 remainders to cycle through: 0,1,2,3,4,5,6. And in fact, the cycle always had to be less than 7, because if the remainder 0 ever occurs, then the fraction just terminates (which gives cycle length 1), so the remainders can only really cycle through 1,2,3,4,5,6.

Now, the same works for $\frac mn$ for any $m > 0, n > 1$.