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# Take Three from Five

*Did you know ... ?*

Although number theory - the study of the natural numbers - does not typically feature in school curricula it plays a leading role in university at first year and beyond. Having a good grasp of the fundamentals of number theory is useful across all disciplines of mathematics. Moreover, problems in number theory are a great leisure past time as many require only minimal knowledge of mathematical 'content'.
## You may also like

### Adding All Nine

### Doodles

Links to the University of Cambridge website
Links to the NRICH website Home page

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Age 11 to 16

Challenge Level

*Take Three from Five printable sheet*

*This problem builds on* What Numbers Can We Make?

Take a look at the video below.

Will Charlie always find three integers that add up to a multiple of 3?

*If you can't see the video, click below to read a description.*

Charlie invited James and Caroline to give him sets of five integers (whole numbers).

Each time he chose three integers that added together to make a multiple of 3:

TOTAL | ||||||

3 | 6 |
5 |
7 |
2 | 18 | |

7 |
17 |
15 |
8 | 10 | 39 | |

20 | 15 |
6 |
11 | 12 |
33 | |

23 |
16 |
9 |
21 | 36 | 48 | |

99 |
57 |
5 | 72 |
23 | 228 | |

312 |
97 |
445 | 452 |
29 | 861 | |

-1 | -1 |
0 |
1 |
1 | 0 |

Charlie challenged Caroline and James to find a set of five integers that didn't include three that added up to a multiple of 3.

**Can you find a set of five integers that doesn't include three integers that add up to a multiple of 3?
If not, can you provide a convincing argument that you can always find three integers that add up to a multiple of 3?**

You can test sets of five integers using the interactivity below.

Click here for a poster of this problem.

Although number theory - the study of the natural numbers - does not typically feature in school curricula it plays a leading role in university at first year and beyond. Having a good grasp of the fundamentals of number theory is useful across all disciplines of mathematics. Moreover, problems in number theory are a great leisure past time as many require only minimal knowledge of mathematical 'content'.

*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.*

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?