15

Problem 28

1. If you were counting in base four, what number would you say just before you said $100_\text{four}$?
2. What number is one more than $133_\text{four}$?
3. What is the greatest three-digit number that can be written in base four? What numbers come just before and just after that number?

Problem 29

Explain what is wrong with writing $313_\text{two}$ or $28_\text{eight}$.

Problem 30

1. Write out the base three numbers from $1_\text{three}$ to $200_\text{three}$.
2. Write out the base five numbers from $1_\text{five}$ to $100_\text{five}$.
3. Write the four base six numbers that come after $154_\text{six}$.

Problem 31

Convert each base ten number to a base four number. Explain how you did it.

\[ 13, \qquad\qquad 8,\qquad\qquad 24,\qquad\qquad 49 \]

Challenges:

\[ 0.125, \qquad\qquad 0.11111\dots = 0.\overline{1} \]

Problem 32

In order to use base sixteen, we need sixteen digits — they will represent the numbers zero through fifteen.  We can use our usual digits 0–9, but we need new symbols to represent the digits ten, eleven, twelve, thirteen, fourteen, and fifteen.  Here’s one standard convention:

1. Convert these numbers from base sixteen to base ten, and show your work:
   \[ 6\textrm{D}_{\text{sixteen}}
   \qquad \quad
   \textrm{AE}_{\text{sixteen}}
   \qquad\quad
   9\textrm{C}_{\text{sixteen}}
   \qquad\quad
   2\textrm{B}_{\text{sixteen}} \]
2. Convert these numbers from base ten to base sixteen, and show your work:
   \[97
   \qquad \quad
   144
   \qquad\quad
   203
   \qquad\quad
   890 \]

Problem 33

How many different symbols would you need for a base twenty-five system? Justify your answer.

Problem 34

All of the following numbers are multiples of three.

\[3, \quad 6, \quad 9, \quad 12, \quad 21, \quad 27, \quad 33, \quad 60, \quad 81, \quad 99.\]

1. Identify the powers of 3 in the list. Justify your answer.
2. Write each of the numbers above in base three.
3. In base three: how can you recognize a multiple of 3?  Explain your answer.
4. In base three: how can you recognize a power of 3? Explain your answer.

Problem 35

All of the following numbers are multiples of five.

\[5, \quad 10, \quad 15, \quad 25, \quad 55, \quad 75, \quad 100, \quad 125, \quad 625, \quad 1000.\]

1. Identify the powers of 5 in the list. Justify your answer.
2. Write each of the numbers above in base five.
3. In base five: how can you recognize a multiple of 5?  Explain your answer.
4. In base five: how can you recognize a power of 5? Explain your answer.

Problem 36

Convert each number to the given base.

1. $395_\text{ten}$ into base eight.
2. $52_\text{ten}$ into base two.
3. $743_\text{ten}$ into base five.

Problem 37

What bases makes theses equations true? Justify your answers.

1. $35 = 120_{\underline{\quad}}$
2. $41_{\text{six}} = 27_{\underline{\quad}}$
3. $ 52_{\text{seven}} = 34_{\underline{\quad}}$

Problem 38

What bases makes theses equations true? Justify your answers.

1. $32 = 44_{\underline{\quad}} $
2. $57_{\text{eight}} = 10_{\underline{\quad}}$
3. $ 31_{\text{four}} = 11_{\underline{\quad}}$
4. $15_x = 30_y  $

Problem 39

1. Find a base ten number that is twice the product of its two digits. Is there more than one answer? Justify what you say.
2. Can you solve this problem in any base other than ten?

Problem 40

1. I have a four-digit number written in base ten. When I multiply my number by four, the digits get reversed.  Find the number.
2. Can you solve this problem in any base other than ten?

Problem 41

Convert each base four number to a base ten number. Explain how you did it.

\[ 13_{\text{four}}
\qquad
322_{\text{four}}
\qquad
101_{\text{four}}
\qquad
1300_{\text{four}}\]

Challenges:

\[ 0.2_{\text{four}}
\qquad\qquad
0.111…_{\text{four}}=0.\overline{1}_{\text{four}}\]

Problem 42

Consider this base ten number (I got this by writing the numbers from 1 to 60 in order next to one another):

\[12345678910111213\ldots57585960.\]

1. What is the largest number that can be produced by erasing one hundred digits of the number? (When you erase a digit it goes away. For example, if you start with the number 12345 and erase the middle digit, you produce the number 1245.) How do you know you got the largest possible number?
2. What is the smallest number that can be produced by erasing one hundred digits of the number? How do you know you got the smallest possible number?

Problem 43

Can you find two different numbers (not necessarily single digits!) $a$ and $b$ so that $a_b = b_a$? Can you find more than one solution? Justify your answers.