23) Consider the fractions with positive numerator and denominator. Which of the following sets is finite?

(A) the set of fractions less than 100 with numerator less than 100

(B) the set of fractions greater than 100 with denominator greater than 100

(C) the set of fractions less than 100 with denominator less than 100

(D) the set of fractions less than 100 with numerator greater than 100

(E) the set of fractions greater than 100 with denominator less than 100.
23) The answer is $(\mathbf{C})$.

Indeed, if $\frac{a}{b}<100$ and $b<100$ it is necessary to have $a<100 \cdot b<10000$ and thus the fractions $\frac{a}{b}$ that satisfy the required conditions are necessarily less than $100 \cdot 10000=1000000$.

It is not difficult to show that in all other cases there are infinitely many fractions: for case (A) the fractions $\frac{1}{n}(n=1,2, \ldots)$ satisfy the required conditions; for case (B) consider fractions of the type $100 n+\frac{1}{101}$, i.e., $\frac{10100 n+1}{101}$; for case (D) to $\frac{100+n}{2 n}=\frac{50}{n}+\frac{1}{2}$ and for case (E) to $\frac{100+n}{1}$.