New divisibility rule! (30,000 of them)

CORRECTIONS
- None yet, let me know if you spot anything!

Filming and editing by Alex Genn-Bash
Written and performed by Matt Parker
Produced and random numbers by Nicole Jacobus
Music by Howard Carter
Design by Simon Wright and Adam Robinson
Dice on loan from the Bec Hill Collection

MATT PARKER: Stand-up Mathematician

[scaredyfish]

I like to imagine Terrible Python is a version of Python, and Matt is very good at programming it.

[JavierMD]

Someone's about to make this code a more efficient

[Paint_The_Future]

That Wikipedia article is about to get a lot bigger.

[thomasschmid]

My favourite divisibility test for 7 is removing multiples of 1001 (=7*11*13). For the number 2716, you can subtract 2002 and get 714, which is obviously divisible by 7.
Works great to get big numbers small by subtracting 10010, 100100 and so on.

[st]

I don't usually comment on videos, but I had to share this story on this video. I went to grade school with a math whiz who worked out a divisibility test for seven, on his own, in 5th grade (I'm sure it was something along the lines of those in this video, but I'm equally sure he worked it out independently; I wish I could remember what it was). He was as proud as I ever saw him when he showed it to the teacher. The teacher's response was "that's neat, but it's kind of like doing cartwheels to get into your chair when you could just walk; why don't you just do the problems the way I showed you." It wasn't even me and it still hurts to think about.

[Oaisus]

My favorite method to see if a number is divisible by 7 is to use a calculator and see if it returns a whole number

[jackdog]

As a programmer, my favourite divisibility test is:
return x%n == 0

[bagelnine]

Matt: "We are putting the prime numbers on ebay."
Also Matt: *holds up the number 9*

[CyberKirby]

"Divisibility Rules" is the motivational phrase that keeps the composites going.

[bencemagasi]

If I'm unable to sleep, then I think of a sufficiently large number, but not too large, to check if it's prime. I take an approximate square root in my head, and then I do the check of divisibility for each prime number up until the biggest one less than the original number's square root.

Because of this, I had to come up with divisibility rules in my head, and I always used this "take away the last digit, multiply it by a correct number, then subtract or add it to the rest" method. This way, I checked that 8329 is indeed a prime number, thus, it's my favorite one!

Coming up with the multipliers was a long mental process, though. I'm glad you made this video, Matt!

[catcoder]

The highest prime test to have been assigned is 23027 (The Taekiro tests). As of the video being released, 23029 and higher are all free real estate! (Coincidentally cut off is between twin primes)

[georgelionon]

This is so base 10 focused. To test if its devisable by a number x, first convert to base x, and if it ends with a zero, its dividable by that..

[jacobstromgren]

I think I prefer my own divsibility by n method...
1. Write the number down.
2. Stare at it for ten seconds.
3. Get annoyed that I can't remember any of the divisibility rules I've written down.
4. Give up.
5. Cave in and get a calculator for the actual result.
Works like a charm every time. 😂

[Uuugggg]

@ 3:08 If you actually continue with 35:
25 + 3 = 28 =>
40 + 2 = 42 =>
10 + 4 = 14 =>
20 + 1 = 21 =>
5 + 2 = 7.
7 is divisible by 7! And 7*5 = 35 where this started.

[craigfjay]

I think fairly early on, it’s easier to use the divisibility rule called “long division”. The only multiplying you need to do is working out the N times table up to 9N, and then it works for all values of N the same way. It even tells you *how many* times your number is divisible by N! (That’s just an exclamation mark, not N factorial)

[Becky_Cooling]

Terrible Python Code is the real star of this show

[dataandcolours]

I am a little surprise the 1001-trick wasn't mentioned. You capture three flies in one bang and it is fast for larger numbers since we don't do operations for every single digit, but instead just every third digit. Since 7•11•13=1001 you can actually check divisibility by both 7, 11 and 13 in one nice swoop by utilizing this. Which is extra cool since they are three consecutive primes!

Here is how you do it. Example 1: I wonder if 11250057 is divisible by either 7, 11 or 13.

1. Group the digit in groups of 3 (hey! That's how we normally write them anyway). So 11 250 057
2. Then make every other group negative (it's wise to choose so we get a positive sum but not necessary). So here -11+250-57 or simply 250-68 = 182.
3. Now all you have to do is to check whether 182 is divisible by either 7, 11 or 13. We notice 182=7•26=7•2•13.
4. Cool 182 is divisible by both 7 and 13. Hence 11250057 is divisible by both 7 and 13 but not by 11.

Another example 2424737531233.

1. Group the digit in groups of 3 (hey! That's how we normally write them anyway). So 2 424 737 531 233
2. Then make every other group negative (it's wise to choose so we get a positive sum but not necessary). So here 2-424+737-531+233 =
3. Now all you have to do is to check whether 17 is divisible by either 7, 11 or 13. We notice 17 is prime.
4. Cool 2424737531233 is neither divisible by 7, 11 nor 13.

[gcewing]

"I called it the vsause method." We live in a bizarre world where, if Matt comes up with something that actually works, he can't call it the Parker method, because that would make people think there was something wrong with it!

[robertm.]

I remember asking my teacher in 4 the grade what the test for divisibility by 4 was. He looked at me like i had a squid hat on.

[Clyntax]

The explanation for how all this works is so much more easy and beautiful! It all comes down to modulo classes. Take Michael's approach. Split any number into two parts so that the number equals 10a+b. If that is divisible by 7, then 3a+b is as well, which if multiplied with 3 is still divisible by 7: 9a+3b. Subtract that from 10a+b and you get a-2b which gives the same rest as a+5b. And that's his formula. Using those calculations you can get super nice divisibility checks. E.g. for 7: Take the first digit. Multiply with 3. Add the next digit. Multiply with 3, add the next digit... Until you added the last digit.
E.g. 112: 1*3 = 3, +1 = 4, *3 = 12, +2 =14 which is divisible by 7, so 112 is as well.