\section*{Problem 3 - 171013}

If \(a_{0}, a_{1}, \ldots, a_{n-1}, a_{n}\) are natural numbers with \(0 \leq a_{i} \leq 7 (i=0,1, \ldots n)\) and

\[
z=a_{n} \cdot 8^{n}+a_{n-1} \cdot 8^{n-1}+\ldots+a_{1} \cdot 8+a_{0}
\]

then it is said that \(z\) is represented by the digits \(a_{n}, a_{n-1}, \ldots, a_{1}, a_{0}\) in the octal system, and it is written briefly as

\[
z=\left[a_{n} a_{n-1} \ldots a_{1} a_{0}\right]_{8}
\]

For example, the natural number that has the representation 135 in the decimal system is [207] \(]_{8}\) because \(135=2 \cdot 8^{2}+0 \cdot 8+7\) and \(0 \leq 2 ; 0 ; 7 \leq 7\) and \(2>0\).

a) Represent the natural number, whose decimal representation is 214, in the octal system!

b) In the cryptogram

\begin{tabular}{rrrrrr}
{[} & \(\mathrm{E}\) & \(\mathrm{I}\) & \(\mathrm{N}\) & \(\mathrm{S}\) & ]\(_{8}\) \\
+[ & \(\mathrm{E}\) & \(\mathrm{I}\) & \(\mathrm{N}\) & \(\mathrm{S}\) & ]\(_{8}\) \\
\hline\([\mathrm{Z}\) & \(\mathrm{W}\) & \(\mathrm{E}\) & \(\mathrm{I}\) & ]\(_{8}\) \\
& \multicolumn{6}{c}{ SOLUTION }
\end{tabular}
it is required to substitute the same digits for the same letters and different digits for different letters so that natural numbers represented in the octal system result, for which the given addition statement is true.

Give at least one solution to this cryptogram, and show that the given solution has all the required properties!
}

a) The highest power of 8 that is not greater than 214 is \(8^{2} \cdot 2 = 64\). The largest multiple of 64 that is not greater than 214 is \(3 \cdot 64 = 192\). Furthermore, \(214 = 192 + 22\) and \(22 = 2 \cdot 8 + 6\). Thus, we get \(214 = 3 \cdot 8^{2} + 2 \cdot 8 + 6\), and since \(0 \leq 3, 2, 6 \leq 7\) and \(3 > 0\), 2140[326] \(]_{8}\).

b) One solution is, for example,

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1389.jpg?height=142&width=394&top_left_y=817&top_left_x=837)

This solution has the required properties: For the different letters \(E, I, N, S, Z, W\), the different digits \(3, 0, 5, 4, 6, 1\) are used, and the same digits are used for the same letters.

They satisfy the conditions \(0 \leq a_{i} \leq 7\) and the leading digits 3, 6 are not 0. The numbers represented are

\[
\begin{aligned}
& {[3054]_{8} = 3 \cdot 8^{3} + 0 \cdot 8^{2} + 5 \cdot 8 + 4 = 1580} \\
& {[6130]_{8} = 6 \cdot 8^{3} + 1 \cdot 8^{2} + 3 \cdot 8 + 0 = 3160}
\end{aligned}
\]

and for them, \(1580 + 1580 = 3160\) is a true statement.