7.3: Unusual Number systems

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May 18, 2022
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- 7.2: Number Bases
- 7.E: Exercises

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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.

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Dualtown Number System

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