## SUBJECT 3

Let the sets $A=\{5 x+3 ; 36\}$ and $B=\left\{x^{2} ; 7 y+5\right\}$, where $x, y \in N$.

a) Determine $x, y \in N$ for which the sets are equal;

b) Determine whether $A \cup B$ can have three elements.
## Subject 3.

a) $u(5 x+3) \in\{3 ; 8\}$ this cannot be a perfect square, so $x^{2}=36$, hence $x=6 \ldots . .2 \mathbf{p}$

![](https://cdn.mathpix.com/cropped/2024_06_07_514927207678d9ecd2fbg-2.jpg?height=68&width=1506&top_left_y=1708&top_left_x=237)

b) For $A \cup B$ to have three elements, $A \cap B$ must have one element, and according to a), it follows that $x=6$

$1 p$
the condition $7 y+5 \neq 33$ must be imposed ..... 1p
thus, for any $y$ natural number different from 4, the union has 3 elements..... ..... $1 p$