
N3. Find all ordered pairs $(a, b)$ of positive integers for which the numbers $\frac{a^{3} b-1}{a+1}$ and $\frac{b^{3} a+1}{b-1}$ are positive integers.


Solution. As $a^{3} b-1=b\left(a^{3}+1\right)-(b+1)$ and $a+1 \mid a^{3}+1$, we have $a+1 \mid b+1$.

As $b^{3} a+1=a\left(b^{3}-1\right)+(a+1)$ and $b-1 \mid b^{3}-1$, we have $b-1 \mid a+1$.

So $b-1 \mid b+1$ and hence $b-1 \mid 2$.

- If $b=2$, then $a+1 \mid b+1=3$ gives $a=2$. Hence $(a, b)=(2,2)$ is the only solution in this case.
- If $b=3$, then $a+1 \mid b+1=4$ gives $a=1$ or $a=3$. Hence $(a, b)=(1,3)$ and $(3,3)$ are the only solutions in this case.

To summarize, $(a, b)=(1,3),(2,2)$ and $(3,3)$ are the only solutions.
$\mathbf{N 4}$. A rectangle in the $x y$-plane is called latticed if all its vertices have integer coordinates.

a. Find a latticed rectangle with area 2013 whose sides are not parallel to the axes.

b. Show that if a latticed rectangle has area 2011, then its sides are parallel to the axes.
