7.E: Exercises

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May 18, 2022
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- 7.3: Unusual Number systems
- Back Matter

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Exercise $\PageIndex{1}$

Convert $101101101_2$ to base $10$ .

Answer: TBD.

Exercise $\PageIndex{2}$

The tripletown number system use only numbers which are $1$ more than some multiples of $3$ .

Construct a $5\times5$ tripletown number system of multiplication table with the numbers $1,4,7,10$ and $13.$
Find the smallest ten prime numbers in the tripletown number system.
Find a number with two different tripletown number system prime factorizations.
Does the Prime divisibility Theorem hold for the tripletown number system? Explain.

Exercise $\PageIndex{3}$

The Quadritown number system use only numbers which are $1$ more than some multiples of $4$ .

Construct a $5\times5$ Quadritown number system of multiplication table with the numbers $1,5,9,13$ and $17.$
Find the smallest ten prime numbers in the Quadritown number system.
Find a number with two different Quadritown number system prime factorizations.
Does the Prime divisibility Theorem hold for the Quadritown number system? Explain.

Answer

Construct a 5X5 Quadritown multiplication table with the numbers 1, 5, 9, 13 and 17.

X	1	5	9	13	17
1	1	5	9	13	17
5	5	25	45	65	85
9	9	45	81	117	153
13	13	65	117	169	221
17	17	85	153	221	289

2. Find the smallest ten prime numbers in Quadritown.

Answer

5, 9, 13, 17, 21, 29, 33, 37, 41, 49

3. Find a number with two different Quadritown prime factorizations.

A solution: (found by trial and error)

Examples
Product	Prime Factorization #1	Prime Factorization #2
441	(21)(21)	(9)(49)
1089	(33)(33)	(9)(121)
2205	(5)(21)(21)	(5)(9)(49)
3249	(57)(57)	(9)(361)

Does the Prime divisibilty Theorem hold for the Quadritown number system? Explain.

Prime divisibility is defined as follows:

Let p be a prime and let a and b be integers. If p ∣(ab)then p∣a or p∣b.

Thus, prime divisibility does not hold for the Quadritown number system since 21 | 441and 21 | (9)(49), but 21 ∤ 9nor does 21 ∤ 49. Note, this argument could be modified for each of the examples identified in part c.

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X	1	5	9	13	17
1	1	5	9	13	17
5	5	25	45	65	85
9	9	45	81	117	153
13	13	65	117	169	221
17	17	85	153	221	289

X	1	5	9	13	17
1	1	5	9	13	17
5	5	25	45	65	85
9	9	45	81	117	153
13	13	65	117	169	221
17	17	85	153	221	289

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X	1	5	9	13	17
1	1	5	9	13	17
5	5	25	45	65	85
9	9	45	81	117	153
13	13	65	117	169	221
17	17	85	153	221	289