7.3: Unusual Number systems
- Page ID
- 7327
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Dualtown Number System
The peoples of Dualtown use only numbers which are 1 and some of the multiples of 2 (even numbers). \(E=\{1, 2, 4, 6,8 \cdots\}\). Notice that the set \(E\) is closed under multiplication.
Let's construct a \( 4 \times 4\) multiplication table \(E\).
\(\times\) | \(1\) | \(2\) | \(4\) | \(6\) |
\(1\) | \(1\) | \(2\) | \(4\) | \(6\) |
\(2\) | \(2\) | \(4\) | \(8\) | \(12\) |
\(4\) | \(4\) | \(8\) | \(16\) | \(24\) |
\(6\) | \(6\) | \(12\) | \(24\) | \(36\) |
The smallest ten prime numbers in Dualtown are \(2, 6, 10, 14, 18, 22, 26, 30, 34, 38\). Notice that \(36=(6)(6)=(2)(18)\). Thus \(36\) has two different Dualtown prime factorizations. Hence the Prime Divisibility Theorem does not hold for the Dualtown number system.