Solution: The problem of the corrupt politicians

The solution to the puzzle “The problem of the corrupt politicians” is the following:

If \(x\) is the number of stolen banknotes, the conditions of the problem are:

$$
\begin{cases}
x = 4x_1 + 1 \\
3x_1 = 4x_2 + 1 \\
3x_2 = 4x_3 + 1 \\
3x_3 = 4x_4 \\
3x_4 = 4x_5
\end{cases},
$$
where \(x_1, x_2, x_3, x_4\) are respectively the number of banknotes that each one takes away as he gets up, and \(x_5\) is the number of banknotes left in the distribution in the morning after.
By subtracting \(x_1\) in the second equation and substituting in the first one,
$$
x=4\Big(\frac{4}{3}x_2+\frac{1}{3} \Big)+1 = 4 \frac{4}{3}x_2+\frac{4}{3} +1
$$
Doing the same with the successive equations gives the following results
$$
x=4\Big(\frac{4}{3}\Big)^4x_5+\Big(\frac{4}{3}\Big)^2 + \frac{4}{3}+1,
$$
whereof
$$
x=4\frac{256x_5+63}{81}+1.
$$
For \(x\) to be an integer, \(256x_5+63\) must be a multiple of \(81\), that is,
$$
256x_5+63=81K,
$$
then
$$
256x_5=81K-63=9(9K-7)
$$
Since \(256\) is prime with \(9\), then \(256\) is a divisor of \(9K-7\). The first multiples of \(256\) are \(256\), \(512\), \(768\), \(1024\), \(1280 \ldots\) The first multiple of \(9K-7\) is \(1280\).
Therefore
$$
256 \cdot 5 = 9 \cdot 143 – 7, \qquad 256 \cdot 45 = 9(9 \cdot 143 – 7).
$$
Thus,
$$
x_5 = 45, x_4 = 60, x_3 = 80, x_2 = 107, x_1 = 143, x = 573.
$$
They stole \(573\) notes and the distribution went like this:
The first one took
$$
x_1+x_5=188 \text{ notes,}
$$
the second one took
$$
x_2+x_5=152 \text{ notes,}
$$
the third one took
$$
x_3+x_5=125 \text{ notes,}
$$
and the fourth one took
$$
x_4+x_5=105 \text{ notes.}
$$
\(3\) notes went into the piggy bank.

 

 (Drawing by Antonio Fraguas – Forges)