As an employee of the Candy Store on Five Drive, you send the store's amazingly famous chewy candy cubes all over the world. But you must pack them exactly as the eccentric owner, Fiver Cuube, has specified.

- Whenever you see 5 single chewy candy cubes, you put them in a
**bar**. - Whenever you see 5 bars, you put them in a
**flat**. - Whenever you see 5 flats, you put them in a
**box**. - Whenever you see 5 boxes, you put them in a
**carton**.

0. Please draw pictures for the above conversions.

1. To keep track of the chewy candy cubes sent to customers, the ordering department uses the form below. Fill in the blanks in this ordering form.

Order | Cartons | Boxes | Flats | Bars | Singles | Total |
---|---|---|---|---|---|---|

1. | 0 | 2 | 1 | 4 | 3 | 298 |

2. | 0 | 2 | 3 | 1 | 4 | 334 |

3. | 0 | 0 | 2 | 4 | 2 | 72 |

4. | 0 | 0 | 4 | 1 | 0 | 105 |

2. Fiver Cuube's head of ordering, Frances Indigo Violet Essen, is one smart cookie (figuratively speaking of course, we deal with chewy candy cubes not cookies!). She has found a way to condense the packaging numbers.

See if you can figure out the new system given in orders 1~4. When you understand the system, fill in the blanks in the orders 5~8.

Order | Packaging | Total | Order | Packaging | Total | |
---|---|---|---|---|---|---|

1. | 2143 | 298 | 5. | 10002 | 627 | |

2. | 2314 | 334 | 6. | 1234 | 194 | |

3. | 242 | 72 | 7. | 443 | 123 | |

4. | 410 | 105 | 8. | 1414 | 234 |

3. Briefly explain how to use the packaging numbers to find the total number of candies.

The right most digit is how many singles, the next digit (moving left) is how many 5's, the next digit is how many 25's, the next digit is how many 125's, and the next digit is how many 625's.

4. Briefly explain how to use the total number of candies to find the packaging number.

Divide the TOTAL by 625 giving you \(N_1\) remainder \(R_1\). \(N_1\) will be your first packing number.

Divide \(R_1\) by 125 giving you \(N_2\) reminder \(R_2\). \(N_2\) will be your second packaging number.

Divide \(R_2\) by 25 giving you \(N_3\) reminder \(R_3\). \(N_3\) will be your third packinging number.

And so on …

5. Fiver Cuube hears about the new system and is impressed, but insists that a little "five" be put on each packaging number to remind customers that the Candy Store is on "Five" Drive. For example, the first packaging number would be \( \bf{2143_\text{five}} \).

As a test, he would like to place thirty orders consisting of 1, 2, 3, … , and 30 total candies. Please give the thirty packaging numbers for Mr. Cuube, in order.

\( \begin{array}{rrrrr} 1_\text{five}& 2_\text{five}& 3_\text{five}& 4_\text{five}& 10_\text{five}\\ 11_\text{five}& 12_\text{five}& 13_\text{five}& 14_\text{five}& 20_\text{five}& \\ 21_\text{five}& 22_\text{five}& 23_\text{five}& 24_\text{five}& 30_\text{five}\\ 31_\text{five}& 32_\text{five}& 33_\text{five}& 34_\text{five}& 40_\text{five} \\ 41_\text{five}& 42_\text{five}& 43_\text{five}& 44_\text{five}& 100_\text{five} \\ 101_\text{five}& 102_\text{five}& 103_\text{five}& 104_\text{five}& 110_\text{five} \end{array}\)