Simple divisibility rules – first explanations
It’s easy to dismiss the divisibility rules for 2, 5, and 10 as “obvious,” but let’s pause for a moment to look at them.
A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.
First, let’s agree that 
A detail. If 2 goes into two numbers, it goes into their sum. Think, 2 goes into 4 twice, and into 6 three times, so it goes into 4+6=10 exactly 2+3=5 times. I should make a picture.
And one more detail, if 10 is divisible by 2, then any multiple of 10 (write 10k or 10n) is divisible by 2.
So now let’s look at some big ugly number: 123456. This is 123450 + 6. 123450 is a multiple of 10, so 2 goes in, and we already agreed about 6. Since 6 goes into both, it goes into their sum, 123456.
This works for any number at all: #$%**1 can be rewritten as #$%**0 + 1. Two goes into the ugly multiple of 10, but not into the 1, so it does not go into #$%**1.
The rule for 5 will work almost the same. 0 and 5 are multiples of 5, and 10 is a multiple of 5, so xyz5 can be rewritten as the sum of xyz0, a multiple of 10, and 5. Since both addends are divisible by 5, so is their sum.
The rules for 4 and 25 are easy extensions. Neither 4 nor 25 go into 10 evenly, but they both go into 100. Rewrite a large number, eg 1234 as 1200 + 34. We then know that 1200, a multiple of 100, is divisible by 4, so we only need to check 34 (nope, 4 doesn’t go into 34, and it doesn’t go into 1234).
25 works the same way, but as there are only 4 two-digit endings (00, 25, 50, 75) it’s even easier to check.
Let’s push on, just a bit. Aren’t 4 and 25 the squares of 2 and 5? And they require checking the last two digits?
What about the cubes of 2 and 5? Well, 8 and 125, neither divides 100, but both divide 1000. So for those two, we could check the last 3 digits.
Why is this working so nicely? hmm. Let’s factor 1000: 
Note: the language here needs lots of work, and I may have flubbed a few mathy details. I’d like this to be accessible to kids and people who don’t normally read math. I am willing to compromise language, but not to the point of being wrong or unnecessarily hazy. If you see something that could use fixing, please, say something. I’d appreciate it.
JD2718 
Hey, some of those LaTex pastes didn’t work in RSS, but on the post they are fine.
When I was 12 or 13, I found a bunch of old “teacher” copies of textbooks that were being discarded in a dumpster, and fished out the one on 8th grade math. It was filled with all sorts of incredibly cool things that weren’t in MY textbook, and I remember it very fondly as one of the “secret” prized possessions of my youth. (I hid the book fearing that I would get in trouble for looking at the answers!)
Reading the post above, and the entries on your blog makes me feel just the same way I did when I read that book. Your students are very lucky to have you. Thank you for reminding me of something great!
Have you looked at the FAQ at Ask Dr Math?
The links at
http://mathforum.org/dr.math/faq/faq.divisibility.html
are pretty good (that page itself is just the rules but it links to a lot of the explanations).
I am going to copy and save your posts for my grandsons. Your explanations are great. By the way, I find myself looking at license plates and other numbers to see what divisibility rules might work for them. Does this mean I’m crazy?
@Joshua, thanks for the link. I like some of the Dr. Math stuff, but have my own way for other parts (both presentations of rules, and explanations). Good tips as I continue this series. Thanks!
@range, the problem was not the latex, but my failure to hit preview. I edited on the fly, finally getting the tex to work. Your rss reflects the first version. If I’d fixed it before publishing…
@jimsmuse, Great story!
I took a crack at simple here and then here a couple of years ago.
If it helps . . . enjoy!
Nice post. You might be interested in my post about a divisibility rule for 7, 11, and other numbers at
http://polymathematics.typepad.com/polymath/divisibility_test_for_7_1.html
It includes (with explanation) a fairly quick test for 7 and 11 that I’ve only rarely seen on- or offline