Subject III
A natural number of the form $\overline{a b c d}$ is called superb if $4 \cdot \overline{a b}=\overline{c d}$.
a) How many superb numbers exist?

b) Show that any superb number is divisible by 13.
Solution:
a) $4 \cdot 10 \leq 4 \cdot \overline{a b} \leq 99$ 2 points
$10 \leq \overline{a b} \leq 24$ 1 point
There are 15 numbers ..... 1 point
b) $\overline{a b c d}=100 \cdot \overline{a b}+\overline{c d}=100 \cdot \overline{a b}+4 \cdot \overline{a b}=104 \cdot \overline{a b}$ ..... 1 point
$104 \cdot \overline{a b}=13 \cdot 8 \cdot \overline{a b}$ ..... 1 point
Finalization ..... 1 point