# 10 Problem Bank

### Problem 28

- If you were counting in base four, what number would you say just before you said ?
- What number is one more than ?
- 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 or .

### Problem 30

- Write out the base three numbers from to .
- Write out the base five numbers from to .
- Write the four base six numbers that come after .

### Problem 31

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

**Challenges:**

### 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:

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

7 | |

8 | |

9 | |

10 | |

11 | |

12 | |

13 | |

14 | |

15 | |

16 |

- Convert these numbers from base sixteen to base ten, and show your work:
- Convert these numbers from base ten to base sixteen, and show your work:

### 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.

- 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.

### Problem 35

All of the following numbers are multiples of five.

- 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.

### Problem 36

Convert each number to the given base.

- into base eight.
- into base two.
- into base five.

### Problem 37

What bases makes theses equations true? Justify your answers.

### Problem 38

What bases makes theses equations true? Justify your answers.

### Problem 39

- 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?

### Problem 40

- 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?

### Problem 41

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

**Challenges:**

### Problem 42

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

- 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?

### Problem 43

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