7. Determine all pairs of integers ( $a, b$ ), for which $a=\frac{4 b-5}{b-2}$.

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
7. The given expression can be transformed in the following way:

$a=\frac{4 b-5}{b-2}=\frac{4(b-2)+8-5}{b-2}=\frac{4(b-2)+3}{b-2}=\frac{4(b-2)}{b-2}+\frac{3}{b-2}=4+\frac{3}{b-2}$

Now it is easy to see that the number $a$ will be an integer if $\frac{3}{b-2}$ is an integer, i.e., if $b-2$ is a divisor of the number 3, or if $b-2 \in\{1,-1,3,-3\}$.

Thus, there are four possibilities.

For $\mathrm{b}-2=1$ it follows that $\mathrm{b}=3$ and $a=4+3=7$.

For $\mathrm{b}-2=-1$ it follows that $\mathrm{b}=1$ and $a=4-3=1$.

For b-2 $=3$ it follows that $\mathrm{b}=5$ and $a=4+1=5$.

1 POINT

For $\mathrm{b}-2=-3$ it follows that $\mathrm{b}=-1$ and $a=4-1=3$.

1 POINT

The required pairs are $(7,3),(1,1),(5,5)$ and $(3,-1)$.

1 POINT