## Subject II. (20 points)

Aida has a problem:

If she divides the number of candies she has by 11, she has 9 left, and if she divides by 5, she has 2 left. How many candies will she have left if she divides by 55?

Prof. Anca Cristina Hodorogea, ISJ Cluj
## Subject II.

$$
\begin{gathered}
\text { T. with remainder } \Rightarrow\left\{\begin{array} { l } 
{ x = 11 \cdot c _ { 1 } + 9 } \\
{ x = 5 \cdot c _ { 2 } + 2 }
\end{array} \Rightarrow \left\{\begin{array}{l}
x+2=11 \cdot\left(c_{1}+1\right) \\
x+3=5 \cdot\left(c_{2}+1\right)
\end{array}\right.\right. \\
\left\{\begin{array}{l}
10 \cdot x+20=55 \cdot 2 \cdot\left(c_{1}+1\right) \\
11 \cdot x+33=55 \cdot\left(c_{2}+1\right)
\end{array} \Rightarrow x+55=55 \cdot c_{3}+42 \Rightarrow x=55 \cdot\left(c_{3}-1\right)+42 \Rightarrow r=42\right.
\end{gathered}
$$

(10 points)