# 2.7: Problem Bank

- Page ID
- 10360

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

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

- Write out the base three numbers from \(1_{three}\) to \(200_{three}\).
- Write out the base five numbers from \(1_{five}\) to \(100_{five}\).
- Write the four base six numbers that come after \(154_{six}\).

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

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

**Challenges:**

\[0.125, \qquad 0.11111 \cdots = 0. \bar{1}\]

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:

base ten |
base sixteen |
---|---|

7 | $$7_{sixteen}$$ |

8 | $$8_{sixteen}$$ |

9 | $$9_{sixteen}$$ |

10 | $$A_{sixteen}$$ |

11 | $$B_{sixteen}$$ |

12 | $$C_{sixteen}$$ |

13 | $$D_{sixteen}$$ |

14 | $$E_{sixteen}$$ |

15 | $$F_{sixteen}$$ |

16 | $$10_{sixteen}$$ |

- Convert these numbers from base sixteen to base ten, and show your work:
\[6D_{sixteen} \qquad AE_{sixteen} \qquad 9C_{sixteen} \qquad 2B_{sixteen}\]

- Convert these numbers from base ten to base sixteen, and show your work:
\[97 \qquad 144 \qquad 203 \qquad 890\]

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

All of the following numbers are multiples of three.

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

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

All of the following numbers are multiples of five.

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

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

Convert each number to the given base.

- \(395_{ten}\) into base eight.
- \(52_{ten}\) into base two.
- \(743_{ten}\) into base five.

What bases makes theses equations true? Justify your answers.

- $$35 = 120 \_\_\_$$
- $$41_{six} = 27 \_\_\_$$
- $$52_{seven} = 34 \_\_\_$$

What bases makes theses equations true? Justify your answers.

- $$32 = 44\_\_\_$$
- $$57_{eight} = 10 \_\_\_$$
- $$31_{four} = 11 \_\_\_$$
- $$15_{x} = 30_{y}$$

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

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

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

\[13_{four} \quad 322_{four} \quad 101_{four} \quad 1300_{four}\]

**Challenges:**

\[0.2_{four} \qquad 0.111 \ldots_{four} = 0. \bar{1}_{four}\]

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

\[12345678910111213 \ldots 57585960 \ldotp\]

- 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? - 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?

Can you find two different numbers (not necessarily single digits!) and so that \(a_{b} = b_{a}\)? Can you find more than one solution? Justify your answers.