Divisible by 3

who: J2
when: at breakfast
where: at the dining table

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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 12345678987654321
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>

Of 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.

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.