4. a) Determine the natural numbers $a$ and $b$, if $[a, b]-(a, b)=34$.

b) Write the set $A=\{1,2,3, \ldots, 1000\}$ as a union of 500 disjoint subsets, such that the sum of the elements of each subset is a perfect square.

Subject elaborated by prof. Valentina Blendea

## SOLUTIONS

## Grade VI
4. a) Let $d=(a, b)$; then $a=d x, b=d y$, with $x, y \in \mathbb{N},(x, y)=1$, and $[a, b]=d x y$. Clearly, $d$ must be a divisor of 34, i.e., $d \in\{1,2,17,34\}$. Considering each case separately, we obtain the solutions: $\quad(a, b) \in\{(1,35),(5,7),(7,5),(35,1) ;(2,36),(4,18),(18,4),(36,2)$; $(17,51),(51,17) ;(34,68),(68,34)\}$.

b) For example, we can consider the following partition of $A$ :

$$
\{1,224\},\{2,223\}, \ldots,\{112,113\},\{225,1000\},\{226,999\}, \ldots,\{612,613\}
$$