Subject 3. Determine the cardinality of the set $A=\left\{\overline{a b} \left\lvert\, \frac{16}{a^{2}+b} \in \mathbb{N}\right.\right\}$.
Subject 3.

|  | $\frac{16}{a^{2}+b} \in \mathbb{N} \Rightarrow a^{2}+b \mid 16 \Rightarrow a^{2}+b \in\{1,2,4,8,16\}$ | $\mathbf{2 p}$ |
| :--- | :--- | :--- |
|  | $a^{2}+b=16 \Rightarrow a=4$ and $b=0$ or $a=3$ and $b=7 \Rightarrow$ the numbers 40 and 37 | $\mathbf{1 p}$ |
|  | $a^{2}+b=8 \Rightarrow a=2$ and $b=4$ or $a=1$ and $b=7 \Rightarrow$ the numbers 24 and 17 | $\mathbf{1 p}$ |
|  | $a^{2}+b=4 \Rightarrow a=2$ and $b=0$ or $a=1$ and $b=3 \Rightarrow$ the numbers 20 and 13 | $\mathbf{1 p}$ |
|  | $a^{2}+b=2 \Rightarrow a=1$ and $b=1 \Rightarrow$ the number 11 | $\mathbf{1 p}$ |
|  | $a^{2}+b=1 \Rightarrow a=1$ and $b=0 \Rightarrow$ the number 10, so the cardinality of set A is 8 | $\mathbf{1 p}$ |