## Task B-4.4.

In some numerical system, with a base less than 25, the product of a two-digit number with identical digits and its double is equal to 1210 (in the same numerical system). What is the number and in which numerical system?
## Solution.

We seek $x$ and $b$ such that $\overline{x x}_{b} \cdot 2 \overline{x x}_{b}=1210_{b}$.

In addition to the condition that the base $b > 2$.

1 point

The number we are looking for is written as $\overline{x x}_{b}=x b+x$.

Then $(x b+x) \cdot 2(x b+x)=b^{3}+2 b^{2}+b$. 1 point

$2 x^{2}(b+1)^{2}=b(b+1)^{2} \Longrightarrow 2 x^{2}=b$. 1 point

Due to the condition on the base, $x^{2}=4$ or $x^{2}=9$. 1 point

The first solution is the number 22 in the base 8 system.

1 point

The second solution is the number 33 in the base 18 system.

1 point

(Verification: $22_{8} \cdot 44_{8}=18 \cdot 36=648=1210_{8}, 33_{18} \cdot 66_{18}=57 \cdot 114=6498=1210_{18}$.)