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?
Our friend 12345678987654321
This is actually a number J2 has seen before. In reading the Number Devil, we played with a pattern:
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?
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