when: at breakfast

where: at the dining table

We were recently talking about divisibility rules and J2 was particularly excited about testing large numbers for divisibility by 3. He asked me to provide a large one to see if his mother could figure out whether it was a multiple of 3. This was the resulting conversation:

J0: Let's try 12345678987654321Of course, we parents expected to hear that he added the digits and saw that the digit sum was multiple of three. Unfortunately, that test is just a trick for him right now. He doesn't really understand how it works or why. His own approach wasn't a trick, but it was a bit lucky.

P: I think it is a multiple of 3

J2: <excitedly> It is!

J0: how do you know?

J2: well, it is a multiple of 111 and 111 is 37 x 3.

P: <stunned/surprised> How do you know it is a multiple of 111?

<conversation continues>

**Our friend 12345678987654321**

This is actually a number J2 has seen before. In reading the Number Devil, we played with a pattern:

1x1 | = | 1 |

11x11 | = | 121 |

111x111 | = | 12321 |

1,111x1,111 | = | 1234321 |

11,111x11,111 | = | 123454321 |

111,111x111,111 | = | 12345654321 |

1,111,111x1,111,111 | = | 1234567654321 |

11,111,111x11,111,111 | = | 123456787654321 |

111,111,111x111,111,111 | = | 12345678987654321 |

We spent a lot of time talking about these, checking them, and extending the pattern to see what would happen next. From here, it isn't so hard to see that 111 is a factor of 111,111,111 and, of course, everyone knows that 111 is 37 x 3. Right?

**Some food for thought**

- Where does that sum of digits test for divisibility by 3 actually come from?
- Does it work for any other numbers (hint: yes!)?
- Wait, doesn't this mean that order of digits doesn't matter for those factors?
- What is the divisibility test for 11 and why is it almost as cool as the one for 3?
- What is the divisibility test for 7 and how does the name "primitive root" help explain how messy it is?

**Unrelated Pictures**

J1's first zometool project, showing the duality of the dodecahedron and icosahedron! |

J1 tackles the NRICH table and chairs challenge, with TRIO blocks. More on this later? |

A funny looking plant with a pretty flower that caught J1's eye. |

The blog Math Recreation talks about a divisibility rule for 11 in this interesting post.

ReplyDeleteBe warned: even if you read this, there is still an alternative, but related form of test you can find for 11.

If you convert a base 10 number in base 8, then you can use the digital root divisibility test for testing divisibility by 7.

ReplyDeletee.g. 329

which has a base 8 of 511

Then 5 + 1 + 1 = 7

Thus, 329 is divisible by 7

329/7 = 47

Yes, that's a nice idea. I wonder, can you use that test to prove this result from Ben Vitale:

Deletehttps://benvitalenum3ers.wordpress.com/2015/03/25/divisibility-by-7-6n-digit-number/