# 7.E: Exercises

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Exercise $$\PageIndex{1}$$

Convert $$101101101_2$$ to base $$10$$.

TBD.

Exercise $$\PageIndex{2}$$

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

1. Construct a $$5\times5$$ tripletown number system of multiplication table with the numbers $$1,4,7,10$$ and $$13.$$
2. Find the smallest ten prime numbers in the tripletown number system.
3. Find a number with two different tripletown number system prime factorizations.
4. 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 $$4$$.

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

 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

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)

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.

This page titled 7.E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Pamini Thangarajah.