## PROBLEM 2

Consider the non-zero natural numbers $x, y, z$. When $x$ is divided by $y$, the quotient is 4 and the remainder is 3. When $y$ is divided by $z$, the quotient is 5 and the remainder is 4.

a) Show that $x \geq 100$.

b) Determine the numbers $x, z, y$ knowing that $x+y+z=153$.

G.M. nr. 4 E: 14479
```
\begin{aligned}
& y=5 z+4 ; z>4 \text {.................................................. } 5 \text {, points } \\
& y>24 \Rightarrow 4 y>96 \Rightarrow 4 y+3>99 \Leftrightarrow x>99 \ldots \ldots . . . . . . . . . . . . . . .1 \text { point } \\
& \mathrm{x}>99, \mathrm{x} \in \mathbb{N} \Rightarrow \mathrm{x} \geq 100 \text {................................................. point } \\
& \text { b) } \quad(4 \mathrm{y}+3)+\mathrm{y}+\mathrm{z}=153 \text {........................................................ } 1 \text { point } \\
& \begin{array}{c}
5 \mathrm{y}+\mathrm{z}=150 \Rightarrow 5 \cdot(5 \mathrm{z}+4)+\mathrm{z}=150 \Rightarrow \mathrm{z}=5 \text {................... } 1 \text { point } \\
\mathrm{y}=29 ; \mathrm{x}=119 \quad \text {......................................... }
\end{array}
\end{aligned}
```