## Task 2 - 080722

Let $a$ and $b$ be arbitrary natural numbers with $a > b$.

a) Calculate all numbers $x$ for which the sum of $x$ and the product of $a$ and $b$ equals the square of the number $a$!

b) Calculate all numbers $y$ for which the difference between the product of $a$ and $b$ and the number $y$ equals the square of the number $b$!
(a) Suppose there is a number $x$ with the required property, then we have: $a b+x=a^{2}$, from which we obtain $x=a^{2}-a b$.

Thus, at most the number $x=a^{2}-a b=a(a-b)$ can be a solution. Indeed,

$$
a \cdot b+a(a-b)=a b+a^{2}-a b=a^{2}
$$

(b) Suppose there is a number $y$ with the required property, then we have: $a b-y=b^{2}$, from which we obtain $y=a b-b^{2}$.

Thus, at most the number $y=a b-b^{2}=b(a-b)$ can be a solution. Indeed,

$$
a \cdot b-b(a-b)=a b-a b+b^{2}=b^{2}
$$