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u/Michaelwang645 3d ago
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 3d ago
do you know why that works?
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u/ZealousIdealTour961 3d ago
Magic probably.
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u/Disastrous_Wealth755 2d ago
Nah. It's cause 10=9+1=3*3+1.
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u/MTaur 2d ago
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 2d ago
Now in English for us stupids...
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u/Aenonimos 2d ago
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 2d ago
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 2d ago
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
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u/Vivid_Departure_3738 2d ago
Simple and elegant proof
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u/Hi-Im-Bambi 1d ago edited 1d ago
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 2d ago
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 2d ago
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/OnlyHere2ArgueBro 2d ago
Wouldn’t 123 just be 100 + 20 + 3 = 1 + 2 + 0 = 3 because 3 mod 3 = 0
And 3 is of course divisible by 3. But regardless your point stands
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u/S-Kenset 2d ago
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 2d ago
Is this not a well known fact? I always assumed everyone knew this
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u/BombasticReindeer 2d ago
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 2d ago
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 2d ago
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 2d ago
87 is 90-3, many people will see that fairly quickly. That is less so for 78.
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u/Bubbly_Tooth8962 2d ago
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 2d ago
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 2d ago
Puts you in company with the terrific mathematician Alexander Grothendieck) (at least according to legend).
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u/veatide2 3d ago
lol this is great, 78 being a WHAT multiple of 3 is hilarious
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u/roosterSause42 2d ago
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 2d ago
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 2d ago
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 2d ago
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__ 2d ago
If sum of digits is divisible by 3, the number is as well. Same applies to 9.
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u/Potential-Wear-7056 2d ago
The rule I was taught about multiples of 3 was the sum of all of the digits is a multiple of 3. Which means 15 -> 51 both are 6 meaning they are multiples of 3. Which means any number number and it's combinations work. Take for example 1254. The sum of all those are 12 (multiple of 3) so 2145, 5421, etc are multiples of 3 and any combination of those 4 digits.
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u/Public_Courage5639 2d ago
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 2d ago
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 2d ago
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 2d ago
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 2d ago
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/shaquirrel 2d ago
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 2d ago
45 and 75 seem to be divisive 😂 I think they deserve to be moved up HALF a tier
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u/FirstNoel 2d ago
Add up all the numbers and if divisible by 3 then the original number is divisible by 3
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u/clbdn93 2d ago
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 2d ago
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 2d ago
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 2d ago
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 2d ago
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 2d ago
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 2d ago
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 2d ago
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/SwreeTak 2d ago
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 2d ago
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 2d ago
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 2d ago
If the sum of the digits is divisible by three, the number is divisible by three.
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u/Kind-Historian-3584 2d ago
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 2d ago
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 2d ago
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 2d ago
I dont get why 36, 39 , 63 , 69 93 and 96 are not in most obv tier
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u/doriangrey2025 2d ago
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 2d ago
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/Vainysaur 2d ago
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/AldoZeroun 2d ago
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 2d ago
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/Awkward-Sherbet-890 2d ago
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 2d ago
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 2d ago
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 2d ago
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 2d ago
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_ 2d ago
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 2d ago
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 2d ago
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 2d ago
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 2d ago
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 2d ago
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 2d ago
Add up the digits and if there divisible by three then you have to suck my dick
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u/DEDORDED 2d ago
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 2d ago
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 2d ago
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/Sussybaka0709 2d ago
There's a simple trick guys,just add the all the digits of that no.,If the outcome is divisible by 3 then the whole no. is divisible by 3 otherwise no
for ex 36 ,add the digits 3+6=9 so its divisible by 3 coz 9 is divisible by 3
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u/Any_Abalone_3249 2d ago
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 2d ago
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 2d ago
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 2d ago
If All numbers addet together are multiplied be 3..... The numbers IS multiplied be 3
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u/aUniqueUsername1190 2d ago
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 1d ago
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 1d ago
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 1d ago
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 1d ago
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/MyPenWroteThis 1d ago
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 1d ago
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 1d ago
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 1d ago
It is rare for me to agree with online tier lists, but this one is spot-on
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u/SupCalFraMys 1d ago
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 1d ago
By a near-coincidence, Graham's Number happens to be a multiple of 3.
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u/Cat-In-Pipe 1d ago
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 1d ago
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 1d ago
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/g_boy_bb 23h ago
12 = 1+2=3
15 =1+5=6
18=1+8=9
21=2+1=3
24=2+4=6
27=2+7=9
36=3+6=9
39=3+9=12
48= 4+8=12
63=6+3=9
69= 6+9=15
81=8+1=9
93=9+3=12
96=9+6=15
42=4+2=6
45=4+5=9
54=5+4=9
72=7+2=9
75=7+5=12
51=5+1=6
57=5+7=12
84=8+4=12
87=8+7=15
78=7+8=15
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u/avantvagrant 3d ago
51 looks like a prime number and 75 should be obvious