## The Off Topic Topic!

For strange, random (and maybe a little pointless!) messages.
jsreed5
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### Re: The Off Topic Topic!

I was working with some students on divisibility checks a few days ago, and they asked me if there were any easy checks for numbers larger than 11 (I wrote up a check for every number up to 11 here).
So, here are some more checks, for every number up to 25.

Our general strategy will be as follows:
If the number we're checking divisibility by is composite, say 14, we'll check two different (relatively prime) factors of the number--factors we already know the rules for. In our example 14 = 2 × 7, so we'll check the rules for 2 and 7. The exceptions to this are 16 and 25, which are powers of prime numbers and must be handled differently.
If the number we're checking divisibility by is prime, such as 17, we'll use a "separate and recombine" method similar to the approach we used for 7. We'll separate the last digit of our number to test from the other digits, multiply it by some number, and either add or subtract it from the remaining digits (depending on what number we're checking divisibility by). Follow the link to the earlier tests for divisibility to see how the rule specifically works for 7.

So, without further ado, let's get started.

---

12:
If a number is divisible by both 3 and 4, it's divisible by 12.
Example: 168432
1 + 6 + 8 + 4 + 3 + 2 = 24, which is divisible 3, so 168432 is divisible by 3.
32 is divisible by 4, so 168432 is divisible by 4.
Thus 168432 is divisible by 12.

13:
Separate the last digit from the others, multiply that digit by 4, and add it to the remaining digits. Keep doing this until you get a small number whose divisibility you can check directly.
Example: 107393
Separate the last digit from the rest: 10739 | 3
Multiply it by 4 and add that to the remaining digits: 10739 + (4 × 3) = 10739 + 12 = 10751.
Do it again: 1075 + (4 × 1) = 1075 + 4 = 1079
Another round: 107 + (4 × 9) = 107 + 36 = 143
One more time: 14 + (4 × 3) = 14 + 12 = 26
26 is divisible by 13, so 107393 is divisible by 13.

14:
If a number is divisible by both 2 and 7, it's divisible by 14.
Example: 141624
141624 is even, so it's divisible by 2.
14162 - (2 × 4) = 14154, 1415 - (2 × 4) = 1407, 140 - (2 × 7) = 126, 12 - (2 × 6) = 0. 0 is divisible by 7, so 141624 is divisible by 7.
Thus 141624 is divisible by 14.

15:
If a number is divisible by both 3 and 5, it's divisible by 15.
Example: 493815
4 + 9 + 3 + 8 + 1 + 5 = 30, which is divisible by 3, so 493815 is divisible by 3.
493815 ends in 5, so it's divisible by 5.
This 493815 is divisible by 15.

16:
Throw out all the digits except the last four. If the thousands digit is even, throw it out; if it's odd, add 8 to the other three digits and then throw out the thousands digit.
Separate the hundreds digit from the other digits, multiply it by 4, and add it to the other number. If the result is divisible by 16, the original number is divisible by 16.
Example: 949312
Throw out all but the last four digits: 9312
The thousands digit is odd, so add 8: 9320
Throw out the thousands digit: 320
Separate the hundreds digit: 3 | 20
Multiple 3 by 4 and add it to 20: (4 × 3) + 20 = 12 + 20 = 32
32 is divisible by 16, so 949312 is divisible by 16.

17:
Separate the last digit from the others, multiply that digit by 5, and subtract it from the remaining digits. Keep doing this until you get a small number whose divisibility you can check directly.
Example: 534803
Separate the last digit from the rest: 53480 | 3
Multiply it by 5 and subtract that from the remaining digits: 53480 - (5 × 3) = 53480 - 15 = 53465
Do it again: 5346 - (5 × 5) = 5346 - 25 = 5321
Another round: 532 - (5 × 1) = 532 - 5 = 527
One more time: 52 - (5 × 7) = 52 - 35 = 17
17 is divisible by 17, so 534803 is divisible by 17.

18:
If the number is divisible by both 2 and 9, it's divisible by 18.
Example: 114282
114282 is even, so it's divisible by 2.
1 + 1 + 4 + 2 + 8 + 2 = 18, which is divisible by 9, so 114282 is divisible by 9.
Thus 114282 is divisible by 18.

19:
Separate the last digit from the others, multiply that digit by 2, and add it to the remaining digits. Keep doing this until you get a small number whose divisibility you can check directly.
Example: 890701
Separate the last digit from the rest: 89070 | 1
Multiply it by 2 and add that to the remaining digits: 89070 + (2 × 1) = 89070 + 2 = 89072
Do it again: 8907 + (2 × 2) = 8907 + 4 = 8911
Another round: 891 + (2 × 1) = 891 + 2 = 893
One more time: 89 + (2 × 3) = 89 + 6 = 95
95 is divisible by 19, so 890701 is divisible by 19.

20:
If the last two digits are divisible by 20, the original number is divisible by 20.
Example: 792860
60 is divisible by 20, so 792860 is divisible by 20.

21:
If the number is divisible by both 3 and 7, it's divisible by 21.
Example: 202167
2 + 0 + 2 + 1 + 6 + 7 = 18, which is divisible by 3, so 202167 is divisible by 3.
20216 - (2 × 7) = 20202, 2020 - (2 × 2) = 2016, 201 - (2 × 6) = 189, 18 - (2 × 9) = 0. 0 is divisible by 7, so 202167 is divisible by 7.
Thus 202167 is divisible by 21.

22:
If the number is divisible by both 2 and 11, it's divisible by 22.
Example: 492514
492514 is even, so it's divisible by 2.
4 - 9 + 2 -5 + 1 - 4 = -11. -11 is divisible by 11, so 492514 is divisible by 11.
Thus 492514 is divisible by 22.

23:
Separate the last digit from the others, multiply that digit by 7, and add it to the remaining digits. Keep doing this until you get a small number whose divisibility you can check directly.
Example: 839017
Separate the last digit from the rest: 83901 | 7
Multiply it by 7 and add that to the remaining digits: 83901 + (7 × 7) = 83901 + 49 = 83950
Do it again: 8395 + (7 × 0) = 8395 + 0 = 8395
Another round: 839 + (7 × 5) = 839 + 35 = 874
One more time: 87 + (7 × 4) = 87 + 28 = 115
115 is divisible by 23, so 839017 is divisible by 23.

24:
If the number is divisible by both 3 and 8, it's divisible by 24.
Example: 962568
9 + 6 + 2 + 5 + 6 + 8 = 36, which is divisible by 3, so 962568 is divisible by 3.
568 ÷ 2 = 284. 84 is divisible by 4, so 962568 is divisible by 8.
Thus 962568 is divisible by 24.

25:
If the last two digits are divisible by 25, the original number is divisible by 25.
Example: 728175
75 is divisible by 25, so 728175 is divisible by 25.

---

If you want some more information on the "separate and recombine" method, this note gives algorithms for every prime number less than 50. And for the theoretically-minded, this page has some information on why the method works.

Have fun, FG. Last edited by jsreed5 on Tue 11th Nov 2014, edited 1 time in total.
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Felina
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### Re: The Off Topic Topic!

FLIP PHONEEEEEEEEEEEES! For the win!
I emit a chaos field which distorts time and space and creates chaos everywhere around me. That's why my room is always a mess. ♫☂♫ ~Let's go dancing and singing in the rain~ ♫☂♫  sunshine, lollipops, and rainbows... Magic
I'm not hiding anything :p except SECRETS...
Shh... SECRET makes a woman, woman

I never understood a reason WHY to be serious??? if weirdness is awesome *~¤~* *-°♣° × °♠° × °♥° × °°-* *~¤~*

Amber Root
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### Re: The Off Topic Topic!

If there's a price for rotten judgement. I guess I already won that.

*spins*

I won't say I in love, no chance no way.

Cabaline
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### Re: The Off Topic Topic!

Guys I'm thinking of dying my hair! I think I'm gonna get a dark purple to lilac ombre    I'm always on hand so feel free to message me about anything I also write articles! I also have a twitter too! Posts: 5913
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### Re: The Off Topic Topic!

Cabaline wrote:Guys I'm thinking of dying my hair! I think I'm gonna get a dark purple to lilac ombre Ooo that's sounds really pretty!
Moses Seixas wrote:give bigotry no sanction

jsreed5
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### Re: The Off Topic Topic!

How come I never knew there was a Disney Fairies TV show???
currently in love babby's first research paper
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Cabaline
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### Re: The Off Topic Topic!

Cabaline wrote:Guys I'm thinking of dying my hair! I think I'm gonna get a dark purple to lilac ombre Ooo that's sounds really pretty!
I am so so excited for it I'm getting it done at the end of November as an xmas present to myself    I'm always on hand so feel free to message me about anything I also write articles! I also have a twitter too! Amber Root
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### Re: The Off Topic Topic!

So I went to the dermatologist and I got three biopsies today for melanoma. ;.; and I have to get more later on. Pain.

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### Re: The Off Topic Topic!

Amber Root wrote:So I went to the dermatologist and I got three biopsies today for melanoma. ;.; and I have to get more later on. Pain.
Oh that's not good.
Moses Seixas wrote:give bigotry no sanction

Amber Root
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### Re: The Off Topic Topic!

It's already been one year since my dog died.

jsreed5
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### Re: The Off Topic Topic!

It's always 46.26.
Every single time, it's 46.26.
Never 46.25 and never 46.28. It's always 46.26.

And while we're at it, I can never get anything but a 28.26 either.
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### Re: The Off Topic Topic!

My doll should be here tommorow, I ordered him back in September. I'm still no where done his raglan. I should probaly put a little time into sewing the sleeve tommorow. The mail has been getting here pretty late like nearly six at night sometimes. I open the door thinking it's the pizza guy no it's the mailman and he's got a print I won in a tumblr giveaway.
Moses Seixas wrote:give bigotry no sanction

jsreed5
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### Re: The Off Topic Topic!

Yes, rough drafts are useful.

Especially if you don't really know your subject too well, and you need to figure out how to convert that one recurrence formula into that other recurrence formula, and you're not sure how to show that the function stays bounded even if it doesn't converge, and you have no idea how to actually analyze this bifurcation diagram.
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### Re: The Off Topic Topic!

One of my coworkers at the place I'm interning at has a small index card sized print featuring the American Manual Alphabet, finger spelling for American Sign Launage. I've been looking at for a good minute (I'm on my break mind you). My recent obsession fixation interest in FX's Fargo has spurred my interest to learn another language. *cough* *Russell is babe* *cough* *I really love me some Wrenchers* *cough*
Moses Seixas wrote:give bigotry no sanction

Amber Root
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### Re: The Off Topic Topic!

/be the person your dog thinks you are/ 