## Task B-4.2.

Students wrote posts for Instagram in pairs to invite people to the "Math Evening." The pair that collects the most likes expects a reward. Maja collected 3731, Matko 2754, but they claim that together they have 6705, which makes other pairs suspicious of cheating.

If we allow the possibility that the numbers are not written in the decimal system, could Maja and Matko be right? Explain.
## Solution.

Let $b$ be the base of the number system used by Maja and Marko.

$$
\begin{array}{cc}
3731_{(b)}+2754_{(b)}=6705_{(b)} & 1 \text { point } \\
\left(3 b^{3}+7 b^{2}+3 b+1\right)+\left(2 b^{3}+7 b^{2}+5 b+4\right)=6 b^{3}+7 b^{2}+0 \cdot b+5 & 1 \text { point } \\
b^{3}-7 b^{2}-8 b & \\
b(b-8)(b+1)=0 & 2 \text { points }
\end{array}
$$

The solutions to this equation are $0, -1$, and 8, so the only possible base is 8.

Maja and Marko were adding likes in the octal system.

Note: A student who claims and verifies that the base $b=8$ satisfies the condition, but does not prove that there are no other solutions, gets 3 points.

A student who claims, without verification, that the only solution is $b=8$ gets 1 point.