Thoughts on Divisibility and Counting
I am always in search of good numbers. When I park in a commercial parking lot, I look for a space with a number that is divisible by 3. I like addresses and phone numbers which involve multiples (such as xxx-1696 since 16 x 6 = 96). I enjoy mathematical dates, like 11/11/11, 10/11/12 or 3/5/15.
So the year 2016 has been a good one so far, numerically speaking. In February, I noted 2/4/16 and 2/8/16 and 2/14/16 (you can do the math). And on 2/29/16, otherwise known as “leap year day,” I started thinking about how to determine if a given year is a leap year.
Leap years occur every four years in order to synchronize the calendar with the astronomical seasons. To be precise, they occur in every year that is exactly divisible by 4, however years that are multiples of 100 are NOT leap years, unless they are multiples of 400. And how can one look at a number and determine if it is a multiple of four? Simply examine the tens and the units digits; if that two-digit number is a multiple of four, then the larger number is also a multiple of four. So 2016 is divisible by 4, but 2014 is not.
The rules for divisibility are taught at a variety of math levels. I remember learning the rules for divisibility during a seventh grade math unit on bases other than 10. My children learned them in fifth grade while doing factors, multiples and prime numbers. The Common Core State Standards doesn’t mention them by name, but begins discussion of division in grade 3, and factors, multiples, and primes are contemplated in grades 4 and 6.
In my work with middle school and high school students, I find great variation; some students are skilled at divisibility rules and others are surprised to hear about them. Everyone knows how to tell if a number is divisible by 2 or 5 or 10. Many of my students think multiples of 3 should work the same way (they should end in 3 or 6 or 9) and although they may know the “finger trick” for the first ten multiples of nine, they haven’t considered how to test larger numbers. The rules for divisibility by 3 and 9 rely on the sum of the digits (in contrast to the rules for 2, 4, 5, and 10).
My favorite divisibility rule is the one I learned most recently; while doing KenKen puzzles, I wanted a method other than long division to determine divisibility by 7. The technique is this:
- Chop off the units’ digit and double it.
- Subtract this from the remaining number.
- Continue this process until the result is 1 or 2 digits.
- If the final number is divisible by 7, then the original number was divisible by 7 [Note that 0 is divisible by 7].
For example: 3052
- Take the 2 and double it to get 4.
- 305 – 4 = 301
- Take the 1 and double it to get 2.
- 30 – 2 = 28 which is divisible by 7.
In the same manner, 2016 can be shown to be divisible by 7. So 2016 is a very good year: it is not only a leap year divisible by 4, but is also divisible by 3, 7 and 9.
Can we use technology to help students test for divisibility? On any calculator, we can simply divide each proposed factor. For more efficiency, use the TI-84+ table with the function Y1 = (number)/x and scroll through the table looking for whole number results.
This can be impractical for very large numbers, but on the TI-Nspire there is a factor command which comes to the rescue.
The question of why divisibility rules work is a fruitful exploration for students. For more on proving divisibility rules, see the resources below.
In my search for multiples of 7, I have a new place to look: license plates. In Connecticut, they recently converted the license plate format to two letters followed by five numbers, such as AB-12345. I can check these 5-digit numbers for divisibility by 7 while stopped at a traffic light.
Which leads me to the question of license plate formats and the Fundamental Counting Principle. Connecticut used to have license plates with three numbers followed by three letters: 123-ABC. When those ran out, they briefly used a format with one number, four letters, and finally one number, such as 1ABCD2, at first without a hyphen and later with one inserted after the third character, like 1AG-HJ2. I disliked those plates because they were hard to remember (a 3-character chunk is more memorable to me if it is all letters or all numbers). Thankfully, the new seven-character plates appeared last year even though the sequence hadn’t been exhausted.
How many possible license plates are generated by each of the designs? Multiply the number of ways to select each character to determine the total:
Old style: 123-ABC: 10*10*10*26*26*26
Interim style: 1ABCD2: 10*26*26*26*26*10
New style: AB-12345 26*26*10*10*10*10*10
The interim style, with 45.7 million plates, has 2.6 times as many plates as the old style. The new style has 67.6 million possible plates which is about 3.8 times as many plates as the old style. And with a one-in-seven probability that the number is divisible by 7, there will be about 9,657,143 multiples of seven out there.
Now that’s a lot of license plates! In the meantime, I’m looking forward to the next great date coming soon: 4/9/16. Not only is it a progression of perfect squares (when will that happen again?) but 492016 is a multiple of 7.
NOTES & RESOURCES:
For more on why divisibility rules work, see the following:
- Math Forum: Explaining the Divisibility Rules [link]
- James Tanton: Divisibility Rules Galore! [link]
For more about using fingers to multiply and why the tricks work, see Kolpas, Sidney J., “Let Your Fingers Do the Multiplying”, Mathematics Teacher, 95(4), April 2002.
New resource: (Updated 10/2017) I just came across this problem-based lesson on license plates by John Rowe, using license plates from Australia and the US. Check it out here.