Question 39
Let $a,b,c,d\in\mathbb N$, with $d\ne0$, and define $f:\mathbb N\to\mathbb N$ by
$$
f(n)=\left\lfloor\frac{an+b}{cn+d}\right\rfloor,\qquad n\in\mathbb N,
$$
where $\lfloor x\rfloor$ denotes the greatest integer not exceeding the real number $x$. Here $\mathbb N=\{0,1,2,\ldots\}$.
Prove that the following statements are equivalent:
- **1.** $f$ is surjective.
- **2.** $c=0$, $b\lt d$, and $0\lt a\le d$.
$$
f(n)=\left\lfloor\frac{an+b}{cn+d}\right\rfloor,\qquad n\in\mathbb N,
$$
where $\lfloor x\rfloor$ denotes the greatest integer not exceeding the real number $x$. Here $\mathbb N=\{0,1,2,\ldots\}$.
Prove that the following statements are equivalent:
- **1.** $f$ is surjective.
- **2.** $c=0$, $b\lt d$, and $0\lt a\le d$.