## Task A-4.1.

Determine all natural numbers $b$ for which the equality $11 \cdot 22 \cdot 33=13310$ holds in the number system with base $b$.
## First solution.

Let $b$ be the sought base of the number system.

The equality in the problem is equivalent to the equality

$$
(b+1)(2 b+2)(3 b+3)=b^{4}+3 b^{3}+3 b^{2}+b
$$

We can simplify the expression by factoring out a factor on the left side, and applying the binomial cube formula on the right side:

$$
6(b+1)^{3}=b(b+1)^{3}
$$

Since $b$ cannot be -1, it follows that $b=6$.