What I call as a periodic number is for exemple
$$0.\underbrace{13}_{period}131313...$$ or $$42.\underbrace{465768}_{period}465768465768.$$
So how can we put theses numbers like a integer fractional, i.e. of the form $\frac{a}{b}$ with $a,b\in\mathbb Z$ ?
To illustrate the method lets take 0.13131313131313131313... as an example
Let $x = 0.13131313131313131313...$
We now multiply by a suitable power of 10 such that the fractional part is the same. In this case 100
$100x = 13.13131313131313131313... = 13 + x$
Thus $99x = 13 \Rightarrow x = \dfrac{13}{99}$
For your second example you need to multiply by a bigger power of 10 but the method is identical.
Adding to the existing answers, note that if your number isn't quite in the right form, you can get it that way easily; so if you want to know about $N=1.02371717171\cdots$, you can write $$1000N=1023+ 0.717171\cdots$$
Apply the method mentioned in the other answers to write the repeating part as a fraction $$1000N=1023 +\frac ab$$ then isolate $N$ again: $$N=\frac{1023}{1000}+\frac a{1000b}$$
Now you will need to combine as indicated to get a single fraction, but that's easy.
Multiplying your number by a suitable power of $10$ we can make some parts the nummber jump to the left of the decimal point leaving identical fractional part. That is $10^mx$ and $x$ have the same fractional part. SO their difference is an integer $a$: That is $a= (10^k-1)x$, this shows $x$ is a rational number.
