# Divisibility Rules – Winter 2014

Our goal was to figure out divisibility rules and to understand them. The kids listed them in the order that they wanted to solve them. They decided: 2, 4, 8, 3, 6, 9, 7. Later they changed this to 2, 4, 8, 3, 6, 9, 11, 7 by adding 11.

They knew rules for 2, 4, 8 so we focused on why they work. Using 8 as an example, they showed that 1000/8=250 and any multiple of 1000, say a*1000, would be divisible by 8 to get a*250. So we just need to look at the last three digits. They were happy with dividing by 2 three times to test so we left that there.

I’ll note here that I deliberately did not review division rules before this class because I wanted to be enmeshed in it with them. I can’t participate verbally of course, but it is fun to tag along mentally. So, I don’t know if there is a cool way to look at the last three digits and check divisibility by 8 or not. Any kids out there have any ideas?

That brings us to divisibility by 3. They did not know a rule for 3. First they tried to figure it out by just thinking and talking.

After awhile I commented that sometimes I look at examples when I am stuck. They made two or three examples.

A bit later I commented that there weren’t very many examples on the white board. They generated a bunch more examples then, “look!” one of them said, “the sum of the digits is divisible by 3.” Probably.

They moved to why. More digging around then this: 10/3 has remainder 1, 100/3 has remainder 1, 1000/3 has remainder 1, and so on.

A little more work then: abc is a*100+b*10+c. So a+b+c are exactly the remainders (!) and if the sum of the remainders is divisible by 3, then the whole thing is. “Awesome”, I thought, and they were very happy. (For those of you that know of it, notice no mention of modular arithmetic.)

A bit more of the same process and they realized that: 10/9 has remainder 1, 100/9 has remainder 1 and so on. From there, it was a very quick proof that the sum of the digits needs to sum to 9.

For 6, they were happy with using both the rules for 2 and then 3. I did push a bit here. I know that is the classic rule but I couldn’t help wondering if there was something that combined the two. They did point out that the rules for 2 and 3 were different kinds of rules. So, they went on.

That was near the end of our time but one of the kids remembered a rule for divisibility by 11 that sounded really weird so we added that to our list. I helped here by saying that figuring out 7 was going to take all the tools we could develop, so we put 11 before 7 on our list.

They all got up to leave and someone said that they thought there were an infinite number of primes, quickly followed by, “I think I can prove that.” Everyone agreed that there were probably infinitely many primes. So, I added that to our list of possible things to do and we called it a day.

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