## PROBLEM 3

a) Show that $\frac{\overline{x x 9}+\overline{8 x}+7}{\overline{x 9 x}+\overline{x 6}} \in N$, where $x$ is a digit in base 10.

b) Calculate $2014-(2013 \cdot 2012+2013): 2013^{2}$.
Solution:

a)

Calculate the numerator and obtain $100 x+10 x+9+8 \cdot 10+x+7=111 x+96 \ldots . . . . .1$ point

Calculate the denominator and obtain $100 x+9 \cdot 10+x+10 x+6=111 x+96$ 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_68bbee96daead035d7c1g-3.jpg?height=131&width=1199&top_left_y=757&top_left_x=243)

b)

Factoring out the common factor from the parentheses 2014 - 2013(2012+1): $2013^{2}$ 1 point

2014-2013.2013:2013 ${ }^{2}$ $\qquad$ 1 point

Completion $2014-1=2013$ $\qquad$ 1 point