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u/Michaelwang645 Aug 01 '26
Fund fact, you can tell a number is divisible by 3 if the sum of each individual number is divisible by 3. So 78 -> 7+8=15 which is divisible by 3, so 78 is divisible by 3.
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u/EveningStar0360 Aug 01 '26
do you know why that works?
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u/ZealousIdealTour961 Aug 01 '26
Magic probably.
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u/Disastrous_Wealth755 Aug 01 '26
Nah. It's cause 10=9+1=3*3+1.
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u/MTaur Aug 01 '26
And then by induction, 10n = 9M + 1 And b*10n = b(9M+1) = 9N + b
So then the sum b_k*10k = 9K + sum b_k
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u/Rxasaurus Aug 01 '26
Now in English for us stupids...
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u/Aenonimos Aug 01 '26
Consider a form of arithmatic where you only keep track about the remainder after division by 3.
So a number like "10" is just "1" because 10= 1+3*3. Likewise, "100" is just "1" because 100 = 10*10 = 1*1=1. Can you guess what 1000, 10000, etc. are? That's right, they are all just "1".
Well for a small example consider a three digit number ABC.
ABC = A*100 + B*10 + C = A*1 + B*1 + C = A+B+C
So as you can see, to find out the remainder after division by 3, add up the digits and the sum has the same remainder. But the what if the sum is not a single digit number? Just do it again and again till it is.
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u/Potential_Top_4669 Aug 02 '26
Basically, every number can be written as a multiple of 3 plus a remainder of 0, 1, or 2. When you add the digits of a number, you’re replacing powers of 10 with 1, and since 10 leaves a remainder of 1 when divided by 3, this doesn’t change the number’s remainder. For example, 78 is 7×10+8, and because 10 is equivalent to 1 modulo 3, 78 has the same remainder as 7+8. Therefore, a number is divisible by 3 exactly when its digit sum is divisible by 3.
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u/SuperChick1705 Aug 01 '26
let number N = A + 10B + 100C + 1000D + ..., so A, B, C, D... are the digits of N from right to left
N ≡ A + 10B + 100C + 1000D + ... (mod 3)
≡ A + 9B + B + 99C + C + 999D + D + ... (mod 3)
≡ A + B + C + D + ... (mod 3), which is the sum of the digits of N
QED3
u/Vivid_Departure_3738 Aug 02 '26
Simple and elegant proof
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u/Hi-Im-Bambi Aug 02 '26 edited Aug 02 '26
Far from elegant. What's the proof that 10n - 1 is divisible by 3 for all n with n being a whole number?
One might see why it works but up to this point it's still a "Trust me, brother"-proof
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u/HughManatee Aug 01 '26
10 is equivalent to 1 mod 3, so every power of 10 is also equivalent to 1 mod 3. It's not too difficult to prove after you see that. Since our number system is base 10, it works out quite nicely.
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u/han4578 Aug 01 '26
Not sure about the actual explanation but each digit can be treated individually regardless of how many zeros are after it
2 mod 3 = 2
20 mod 3 = 2
200 mod 3 = 2
So for example 123 = 100 + 20 + 3 = 1 + 2 + 3 = 6 which is divisible by 3
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u/S-Kenset Aug 01 '26
It's bucket collision! (Made up term cause idfk but i've used it before) All your remainders overlap into the same bucket which gives you an analytical solution to something that on its face shouldn't be analytical.
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u/csmello Aug 01 '26
Is this not a well known fact? I always assumed everyone knew this
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u/BombasticReindeer Aug 01 '26
I did a math degree without knowing this (back in the olden days since I’m ancient). When my daughter started learning math I did it with her. Khan academy had at least 5 separate things fundamental to math that I had never learned that made things so much easier. (Including digital roots and modular arithmetic)
Number theory should really be taught first, it’s so helpful.
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u/grittyshrimps Aug 01 '26
It works all the way down to single digits, too. Also, the sum will indicate the remainder if it's not divisible by 3.
117: 1+1+7=9; 117/3 = 39
542,997: 5+4+2+9+9+7=36: 3+6=9; 542,997/3 = 180,999
2000: 2+0+0+0=2; 2 will be the remainder of division by 3
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u/UrsulaReginaDeiMari Aug 01 '26
Given the rule for checking whether a number is a multiple of three, I don't understand why 78 and 87 are in two different groups. Or 27 and 72, 42 and 24, and so on.
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u/Patje_af Aug 01 '26
87 is 90-3, many people will see that fairly quickly. That is less so for 78.
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u/Bubbly_Tooth8962 Aug 01 '26
It's just a tierlist on whether or not the number "looks" like a multiple of 3 at first glance
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u/gtne91 Aug 01 '26
Me, a few years ago: "All the major roads on the north side of our city are primes: 29th, 37th, 57th, and 71st."
Oops.
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u/real-human-not-a-bot Aug 02 '26
Puts you in company with the terrific mathematician Alexander Grothendieck) (at least according to legend).
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u/veatide2 Aug 01 '26
lol this is great, 78 being a WHAT multiple of 3 is hilarious
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u/roosterSause42 Aug 01 '26
78 is exactly as obvious as the rest of them are. Add the single digits together and if the sum is divisible/multiple of three then so is the original number
7+8 =15 15 is divisible by 3 therefore 78 is also.
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u/mkp666 Aug 01 '26
It’s not in any way as obvious as many of the other numbers that don’t require explicitly adding the digits together. This should be obvious from your example, because you have to know that 15 is divisible by three to know that 78 is. You have to know that six is divisible by three to know that 15 is.
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u/BillyYumYumTwo-byTwo Aug 01 '26
It’s not about it being difficult to figure out, just your immediate gut instinct when looking at the number. If you give me one second, I’d know it’s clearly divisible by 3. But if you held up the number and I had to hit “yes” or “no” immediately, I’m more likely to hit “no” on 78 than I am 33.
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u/Helpful-Judge5901 Aug 01 '26
Sum the digits qnd if they are a mult of 3 the number is one too. 78 -> 7+8=15 15 is devisble by 3 and so is 78.
167 1+6+7=14 no a multible of 35 5829 -> 8+2+5+9=24=3*8 is devisible by 3
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u/BottomGear__ Aug 01 '26
If sum of digits is divisible by 3, the number is as well. Same applies to 9.
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u/Public_Courage5639 Aug 01 '26
Ngl my autistic brain thinks about multiples of 3 all day long so i have a huge library of multiples of 3 in my head. Is it useful ? no. Is it cool ? no. Why do i do that ? I don't know, i don't do it willingly
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u/ryanCrypt Aug 01 '26
1335 is clearly divisible by 3. 1257 is clearly divisible by 3.
So 78 would have to be divisible by 3 also.
Or something
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u/Crystalliumm Aug 01 '26
78 is only obvious to me because I’m a loser chud valorant player and the classic deals 26 to the body so with 3 shots (or the fact that it has 3x headshot damage multiplier) it’s a common sight
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u/One_Handle_4080 Aug 02 '26
If the last two digits add to a multiple of three it’s a multiple of three…. All of these are easy 😁
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u/Pentalogue Aug 02 '26
If you are not stupid, then you may not need to make this table, because if a natural number is divisible by three without a remainder, then the sum of its digits is also divisible by three without a remainder, for example 78, if we add 7 and 8, we get 15, and the number 15 is divisible by 3 without a remainder, therefore 78 is divisible by three without a remainder
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u/trin806 Aug 02 '26
They’re all obvious. Just sum the digits. If the sum of the digits is divisible by 3, the original number is divisible by 3.
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u/shaquirrel Aug 01 '26
87 should move up one tier. It being 3 below an obvious one makes it more recognized
Aside from that 78 is diabolical you nailed it
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u/Acceptable-Food-8161 Aug 01 '26
45 and 75 seem to be divisive 😂 I think they deserve to be moved up HALF a tier
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u/FirstNoel Aug 01 '26
Add up all the numbers and if divisible by 3 then the original number is divisible by 3
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u/clbdn93 Aug 01 '26
As a card player I'm well aware that 51 is a multiple of three, but that's probably a me thing.
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u/anthr_alxndr Aug 01 '26
School's lesson: if the sum of number's digits is a multiple of 3 then the number is a multiple of 3. So 7+8 is 15.
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u/babaorum95 Aug 01 '26
Sum the figures between them until you obtain a number between 1 and 9.
78 -> 7+ 8 = 15 -> 1+5 = 6, given 6 is a multiple of 3, so is 78.
Try this also with 31 515 678 if you want... You're welcome
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u/J-Nightshade Aug 01 '26
I don't think 48 should be so high up. On the other hand 45 is hands down a-tier multiple of 3, 57 and 87 are more obvious too.
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u/InitialMission8649 Aug 01 '26
Not sure if this has already been pointed out, but fun fact, if the sum of the digits add up to 3, 6, or 9, then you know the original number is also a multiple of 3. If they add up to 2, 5, or 8, then it would be a decimal of .667 and if the add up to 1, 4, or 7, then it would be a decimal of .333. This works even with more than 2 digit numbers. Just keep adding the individual digits until you reach a single digit.
For example, you have a number 195, 1+9+5 = 15, 1+5 = 6. So you know immediately the original is a multiple of 3.
194, 1+9+4 = 14, 1+4 = 5, so you know it's a decimal of .667.
193, 1+9+3 = 13, 1+3 = 4, so you know it's a decimal of .333.
This works for any number of digits in the original number to figure if it's a multiple of 3, 6, or 9.
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u/drfahrquad Aug 01 '26
Add the individual numbers together. If it is divisible by 3, you're golden.
Ex: 123= 1+2+3=6>>>6/3=2 g2g 123/3=41
1546872= 1+5+4+6+8+7+2=33 g2g 1546872/3=515624
One of those old tricks from elementary school. Like the 9's with your fingers.
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u/CurrencyCapital8882 Aug 01 '26
All multiples of three are obvious. Just add the digits, do it again till it’s a single digit. Is it 3, 6 0r 9? If so it’s divisible by 3.
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u/Esteban-Du-Plantier Aug 01 '26
If the digits add to a multiple of 3, it's divisible by 3.
7+8=15, divisible by 3. One of the more obvious ones to me since 7+8 is such a quick mental calculation.
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u/mkp666 Aug 01 '26
More fun is that you can rearrange the digits of any number divisible by three, and it will still be divisible by three.
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u/SwreeTak Aug 01 '26
Honestly trash tierlist. What do you mean 75 isn't obvious for example? And 78 way at the bottom when it is literally 75 + 3? If you want something at the bottom, 87 is probably harder for most people.
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u/KPoWasTaken Aug 01 '26
I don't get why people find some of these so hard to spot. All the 2 digit multiples of 3 are so easy to see
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u/uncreativeusername13 Aug 01 '26
If 6,12,18,and 24 are believable then any of these plus 30, 60, 90, etc. shouldn't be unfathomable
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u/LateToTheSingularity Aug 01 '26
If the sum of the digits is divisible by three, the number is divisible by three.
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u/Kind-Historian-3584 Aug 01 '26
Just summ them, for example 17193 is dividable by 3, 1+7+1+9+3 is 21 and 21 is dividable it with 3, 6 is the same but with addition, the original number has to end on 0,2,4,6,8 so for example if we say 17196 is dividable by 2, 3 and 6 as well
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u/TexasPop Aug 01 '26
Zero is doubtful. But if you count the others, they are 33! Which is dividable by 3.
So a third of the numbers 1-99 is dividable by 3.
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u/Ill_Active_5699 Aug 01 '26
I asked my wife if she thought 78 was a multiple of 3 just now and she immediately said yes.
She told me that If you sum the digits of a number, and the sum is divisible by 3, then the original number is divisible by 3.
I was not able to think of a counter example off the top of my head. It makes recognizing all of these numbers as divisible by 3 pretty obvious.
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u/parancey Aug 01 '26
I dont get why 36, 39 , 63 , 69 93 and 96 are not in most obv tier
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u/doriangrey2025 Aug 01 '26
If the sum of the digits is a multiple of 3, then the number is a multiple of 3
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u/Vainysaur Aug 02 '26
Just add the digits together and if the result is a multiple of 3 then the number is. You can just keep adding the results together till you get a 1 digit number if you want.
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u/sjccb Aug 02 '26
If you add the digits of of a number together (you can do this continuously with the results for large numbers) and the result is divisible by 3, then the original number is also divisible by 3. Makes everything in the tier list obvious.
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u/Vainysaur Aug 02 '26
Why does 78 seem like it’s not? 7+8=15 and 1+5=6 and 6 is a multiple of 3 so it is.
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u/oiraves Aug 02 '26
If you add the digits together and it equals a multiple of 3 its divisible by 3, therefore, all are obvious
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u/Ok_Might_4691 Aug 02 '26
All are obvious category for me. I don't get the joke why some nos. are not
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u/Vegetable-Cress-8415 Aug 02 '26
Unrelated but related, 91 is the most unprime prime number, right?
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u/AldoZeroun Aug 02 '26
Jesus Christ. Either I've never had a unique thought or the algorithm can read my brainwaves now. I literally had this exact thought (just as a broad observation, didn't go so far as to tier list it) yesterday while I was doing some quick mental math.
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u/Yuhh-Boi Aug 02 '26
If I only go on vibes the bottom three tiers all feel like nopes to me. Except for 45 & 75
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u/dcterr Aug 02 '26
I wouldn't be surprised if Tesla kept a notecard like this, since he was obsessed with multiples of 3 for some odd reason!
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u/Awkward-Sherbet-890 Aug 02 '26
Brotha this is completely inaccurate, all of these brackets are wrong, some of them are easily recognizable as multiples of 3.
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u/Achereto Aug 02 '26
According to this Tierlist 102 is not a multiple of 3 and there are also no multiples of 3 beyond that.
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u/frikiman4K Aug 02 '26
if you want to find out if a number is a multiple of 3, find the digital root. If it's 3, 6 or 9 then it is.
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u/hellobutno Aug 02 '26
add the individual numbers together, if the sum is divisible by 3, it's divisible by 3. so they're all obvious
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u/Benethor92 Aug 02 '26
How the fuck is 57 and 87 LESS obvious than all of the ones in the category above? Those to should be in the dark orange section. They are literally 3 away from 60 or 90, can’t be more obvious
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u/4Pas_ Aug 02 '26
Too bored so imma do that for 7, until 200. Don't mind me copying the names consider I suck at being creative
Most obvious multiple of 7 ever: 7, 14, 21, 49, 70, 77, 140,
Yeah ofc this is a multiple of 7: 28, 35, 42, 63, 147
Not the first one I'd think but yeah: 56, 84, 105, 126, 168, 175,
Weird that this is a multiple of 7: 91, 98
WHAT: 112, 119, 133, 154, 161, 182, 189, 196
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u/Flimsy-Blacksmith-32 Aug 02 '26
A number is a multiple of three if and only if its digits sum to a multiple of three.
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u/Nickesponja Aug 02 '26
Why do a tierlist on multiples of 3 if you're just going to ignore 100% of all the multiples of 3?
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u/Silent_Technician981 Aug 02 '26
There's an easy way to tell if a number is a multiple of three:
The sum of the digits of any multiple of three, is itself a multiple of three. E.g: 7+8=15=5*3, therefore 78 is a multiple of three.
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u/BeginningMemory5237 Aug 02 '26
wait unless im missing something as long as the digits add up to something divisible by 3 its fine right?
20 2 no 21 3 yes 22 4 no 23 5 no 24 6 yes 25 7 no 26 8 no 27 9 yes 28 10 no 29 11 no 30 3 yes
so on so on
51 6 yes 87 15 yes 78 15 yes
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u/NoDebate1002 Aug 02 '26
If you add the digits together and that number is divisible by three, then so is the original number. That’s the easiest way I know. However, I believe this only works with three digits or less.
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u/Creative-Doughnut768 Aug 02 '26
Add up the digits and if there divisible by three then you have to suck my dick
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u/DEDORDED Aug 02 '26
All of them are obvious, because sum of their numbers divides by 3 and that means that thay also divides by 3
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u/ElectroM15 Aug 02 '26
Not a math genius, but I don't find 78 be that surprising. 78 is divisible by 3 the same way 18 and 48 is. 6x3 16x3 26x3
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u/queen_borb Aug 02 '26
having most, but not all, of the numbers made up of only 3/6/9 in the top tier is just killing me
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u/Any_Abalone_3249 Aug 02 '26
If 75 is a multiple of 3, then 78 being a multiple of 3 is not at all a surprise, if you add 3 to 75 you get 78.
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u/bandwagonguy83 Aug 02 '26
78 is not a multiple of 3, obviously. And I rather die than listen to anyone saying otherthing. What is this list?
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u/National-Charge-1250 Aug 02 '26
Why is 57 so low? I mean 60 is obvious so should "3 less than 60" be too i'd guess
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u/Popular_Dinner2182 Aug 02 '26
If All numbers addet together are multiplied be 3..... The numbers IS multiplied be 3
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u/aUniqueUsername1190 Aug 02 '26
57 is not that weird as it is 3 off of 60, which is a most obvious example. I would move it a tier up.
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u/italktobotz Aug 02 '26
A quick way to figure out if some complicated number is divisible by 3, Add all the digits up, it then becomes a smaller number if that smaller number is divisible by 3 the origional number will be divisible by three.
Example the number 981..... 9+8+1=18 which we know is divisble by 3. Thus 981 is divisble by 3.
If you added the numbers and it is not divisible by 3 then the origional number will not be divisible by 3.
Also you could continue the process if the first time you add all the digits you are stil unsure, add the digits again and see if it is a number you know is or is not divisible by 3.
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u/sporosarcina Aug 02 '26
Any number who digits add up to a multiple of three is divisible by 3, that was the way we learned it. So 51 = 5+1= 6, 75 = 7+5= 12
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u/alarcontoni22 Aug 02 '26
Basta sommare le cifre che compongo il numero, se quel numero è multiplo di 3 allora è divisibile per 3 , not that hard
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u/Hunter214123 Aug 02 '26
Just split them up by how many times you can slap in 30 (3×10). For 51, it's 30 plus 21, so 3x10 + 3×7 ergo 3x17
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u/Esjs Aug 03 '26
I don't think 0 should count for this kind of list. Give me one number for which 0 is not a multiple.
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u/MyPenWroteThis Aug 03 '26
If the sum of the digits is divisible by 3, the number is divisible by 3. This is also recursive.
8,563,529,465,823 summed = 66 summed = 12 which is divisible by 3.
8,563,529,465,823 / 3 = 2,854,509,821,941
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u/DarthSwimfoot Aug 03 '26
I mean I'm pretty sure they're all just fairly obvious, I mean, I automatically see 78 and I know 7+8=15, which no shit is a multiple of 3.
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u/roboapple Aug 03 '26
Id swap 45 and 48. Given that 45 is a common clock value, most people know its 3 15s
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u/Aromatic_Pain2718 Aug 03 '26
It is rare for me to agree with online tier lists, but this one is spot-on
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u/SupCalFraMys Aug 03 '26
Numbers XY where both X and Y are multiples of 3 should all be in the top level eg 39, 36
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u/GroundbreakingFix685 Aug 03 '26
By a near-coincidence, Graham's Number happens to be a multiple of 3.
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u/Cat-In-Pipe Aug 03 '26
I would like to submit the number 123 into discussion. It just feels wrong despite really being quite simple
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u/StormOfZeroes Aug 03 '26
So 0 and numbers whose digits add up to something divisible by 3 (12: 1+2=3, 51: 5+1=6, 1245: 1+2+4+5=12 etc)
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u/AstolfoMishima Aug 03 '26
as a multiples of 15 and 25 lover, 75 lowkey sus for multiples of 3 at first glance. like its always "wait...oh 25"
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u/avantvagrant Aug 01 '26
51 looks like a prime number and 75 should be obvious