Is there a way to test parity of fractional part (only period) of irredecible rational number without calculation?

Question

I search in the web to get any way to test parity of fractional part of irredicible rational number by means to know if that fraction (period) even or odd but i didn't get , for example the fraction part of $\frac {17}{19}$ equal to $ \overline{894736842105263157} $ which is odd integer? Any way to test that parity without calculation ? by means how we use the fundemantal arithmitic in number theory to test parity of that period ?

Answer 1

It helps to look at fractions of the form $\frac 1b$ first with $b$ not divisible by $5$. If $b$ is odd, the repeat of the decimal will be odd. If you think about doing the long division to compute the repeat, the last subtraction must result in a remainder of $1$ so you start back around the cycle. That means the number subtracted from $0$ must be odd, so the last digit of the repeat must be odd. We can even say more-the last digit $d$ of the repeat is such that $db\equiv 9 \bmod 10$, so if the last digit of $b$ is $1$, the last digit of $d$ is $9$ and so on. The repeat for $\frac ab$ will just be the repeat for $\frac 1b$ multiplied by $a$, so will be even if $a$ is even and odd with $a$ odd.

If $b$ ends in $5$, we are really interested in $\frac 1{2b}$ because the factor $5$ will just make a leading digit before the repeat starts. Similarly, if $b$ is even, we are interested in $\frac 1{5b}$, using the $5$ to neutralize the factor $2$.