## Subject III. (20 points)

Raluca received on her birthday a sum of money equal to the arithmetic mean of the three-digit natural numbers which, when divided by 5, give a remainder of 2, when divided by 7, give a remainder of 5, and when divided by 8, give a remainder of 1. How much more money does Raluca need to buy a PlayStation 4 console that costs 1799 lei?

prof. Cristian Petru Pop, ISJ Cluj
## Subject III. (20 points)

Let x be the amount of money Raluca has.

$$
\left.\left.\begin{array}{l}
x=5 c_{1}+2 \\
x=7 c_{2}+5 \\
x=8 c_{3}+1
\end{array}\right\} \Rightarrow \begin{array}{r}
x+23=5\left(c_{1}+5\right) \\
x+23=7\left(c_{2}+4\right) \\
x+23=8\left(c_{3}+3\right)
\end{array}\right\} \Rightarrow\left\{\begin{array}{l}
5 \mid (x+23) \\
7 \mid (x+23) \\
8 \mid (x+23)
\end{array}\right.
$$

$\Rightarrow (x+23)$ can be the least common multiple (LCM) of the numbers $5, 7, 8$ or its multiples

$\Rightarrow (x+23) \in \{280, 560, 840\} \Rightarrow x \in \{257, 537, 817\} \Rightarrow M_{a}=537$

Raluca needs 1262 lei more.