889014113748268 mod 222?

I would know how to solve this task which I have copied from the World Mental Calculation Cup that took place in 2018: https://www.recordholders.org/downloads/worldcup/tasks-2018/tasks2018.pdf

Thanks

By the way, I don’t even know what mod or modular arithmetic is, so I ask you to be patient ^^

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It’s the remainder after devision. So if you had 17 mod 7; the answer would be 3, because 2 x 7 = 14 and 17 - 14 = 3. You can also think of it as subtracting the number over and over

17 - 7 = 10
10 - 7 = 3

…since 3 is less than 7 you know you’re done. Your approach depends on the numbers though… the easiest is mod 10 since you’re working in base 10:

123,456 mod 10 = 6

…just walk up by 10, so 20, 30, 40, 50, … 123440, 123450 and done. You can’t go to 123460 because that’s more than what you’re looking for… and 123456 - 123450 = 6. Ultimately though, you’re just moving the decimal point one to the left in base 10 when you divide by 10 and thus you get the 6 after the decimal point as your answer.

Maybe you should first have a look around on the internet to understand this kind of calculation before you get into details as far as the question you’ve looked at with 222. Anything mod 2 will be 0 if the number is even and 1 if the number is odd for example, and that’s a bit fundamental to understanding why you’d take a certain approach to a problem involving modulo math.

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I am curious as to how others like @Kinma would calculate 889014113748268 mod 222.

Do you just divide or do you guys use other methods?

I have to take the division route because I don’t know any methods. 889014113748268 / 222 got me an answer of 4,004,568,079,947 with 34 left over.
I’ve never calculated such a big mod problem before in my head, fortunately this problem isn’t too hard.

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It asks for the remainder of the division, so the straight forward solution would be to use the algorithm for long division. That’s relatively easy to do mentally in this case because you have to keep just a few numbers in working memory.

  1. Take digits from the left of the divisor, 889014113748268, until you have a number bigger than 222. 8 is smaller, 88 is still smaller, but 889 is bigger than 222.

  2. Find the biggest multiple of 222 that is smaller than the result from the previous step, 889. This obviously has to be 888, which is 222 times 4.

  3. Substract the second number from the first one: 889 minus 888 is 1.

  4. Add yet-unused digits from the divisor until the number from the last step is bigger than 222 again. The number from the last step is 1, the next unused digits after 889 are 01411… 10 is smaller than 222, 101 is still smaller than 222, but 1014 is bigger.

The next steps would be to calculate 1014 minus 888, and so on.

\quad 889014113748268\\ -888\\ \qquad 1\\ \qquad 1014\\ \quad \ -888\\ \qquad \ \ 126\\ \qquad \ \ 1261\\ \quad\ -1110\\ \qquad\quad 151\\ \qquad\quad 1511\\ \quad\ \ \ -1332\\ \qquad\quad\ \ 179\\ \qquad\quad\ \ 1793\\ \qquad\ -1776\\ \qquad\quad\quad\ \ 17\\ \qquad\qquad\ \ 1774\\ \qquad\quad\ (-1554)\\ \qquad\qquad\quad 220\\ \qquad\qquad\quad 2208\\ \qquad\quad\ \ \ (-1998)\\ \qquad\qquad\quad\ \ 210\\ \qquad\qquad\quad\ \ 2102\\ \qquad\qquad\ (-1998)\\ \qquad\qquad\qquad 104\\ \qquad\qquad\qquad 1046\\ \qquad\qquad\quad\ -888\\ \qquad\qquad\qquad\ \ 158\\ \qquad\qquad\qquad\ \ 1588\\ \qquad\qquad\quad\ -1554\\ \qquad\qquad\qquad\quad\ \ 34\\

When you are really good in mental math, then you can skip some of the calculations.

  • Take a look at number above the first one in brackets, 1774. You calculated in the previous step that 1776 is a multiple of 222, so you can immediately see that the next result should be 222 minus 2, which is 220.

  • Look at the next number, 2208. You know that 222 times 10 is 2220, which is bigger by 12. Hence the number to follow has to be 222 minus 12. The same applies for 2102, which is 118 smaller than 2220. Hence the next number is 222 minus 118, which is 104.
    Or you see that 2220 times 9 equals 2220 minus 222, which can easily be calculated as 1998. Then for 2208 and 2102 you just have to substract 2000 and then add 2 again.

Just my two cents. I can’t tell if there are some tricks or better shortcuts unfortunately because I don’t do mental math :slight_smile:

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I participated in the contest. I solved that, because it was one of the easiest tasks.

The thing is that you wrote all these intermediate calculations, which are not allowed in the competitions.

We are only allowed to write the final result.

That task was particularly easy because the divisor has the same digits (2,2 and 2) . So, every multiple up to 9, is found instantly 222, 444, 666, 888, 1110, 1332, 1554, 1776, 1998. (at least for world cup level)

So, essentially it is just a dozen of easy subtractions, 3 or 4 digits at a time. Also, it is crucial to remember the zeros.

We had to do around 100 of these tasks in 10 minutes, so it was expected to find this remainder in 6 seconds (in perfect circumstances.)

But most calculators including me, took at least 20 sec, to find such a remainder (mod222) , because one has to be careful and accurate.

For example, if you found 33mod222 or 35mod222 instead of the correct 34mod222, then you get ZERO points (instead of 1 point.). These are the rules, and there is no lenience for getting close to the result. You either find the correct result or not. It is black or white.

Also, potential markings on the paper which indicate intermediate steps, result in instant disqualification. The reason being that it is mental math and not high school math.

We only write down the final result, for practical reasons, just to ensure simultaneous fairness among competitors, because time-wise it would take too long for the judges to make everyone recite the result individually.

In the contest , I think , I found around 20 of such mods. I recall that mod101, mod222 and mod444 were probably the only 3 digit mods that I picked. The others, were small easy ones like 7, 11, 14, 27, 33, 44 and so on, which are also very easy to divide in.

Out of these 100 mods , we could choose to do as many as we could in 10 minutes. And if one found 20, 30, or 40 correct, that was considered a very good result. I don’t think many people found more than 60 out of 100. I will ask Wenzel (who won these surprises), about how many did he found, but I think he was somewhere between 60 and 80 correct. (due to time limitation of 10 minutes).

Bjoern participated also, and maybe he can remember more details , methods or results about that task.

Regards,
Nodas, Greek competitor
2nd place Calendar, MC World Cup 2018

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889014113748268 mod 222.

Here is how I do this.

I focus on 889 and mentally subtract 888.
Remainder 1.
I concatenate ‘014’ to the remainder => 1014.
Subtract 888 and remainder is 126.
Concatenate 1=> 1261.
5*222 = 1110. Subtract that.
Remainder is 151.
=> 1511.

Until here it is more of less the same as Finwing does.
I could subtract 1332, but 1110 forces itself first. So I subtract 1110 first.
Remainder 401.
Then subtract 222.
Remainder 179 => 1793.
Again, I subtract in steps.
The reason is that with each step I want to get rid of the leftmost digit.

In my mind I do:
1793 - 1110 =683 (got rid of the ‘1’)
683 - 666 = 17 (got rid of the ‘6’)
=> 1774.

1774 - 1110 = 664.
664 - 666 = -2
-2 + 222 = 220
220 => 2208
2208 - 2220 = -12 (add 222) => 210 => 2102

2102 - 2220 = -118 (add 222) => 104 (=222-118)
=> 1046.

1046 - 1110 = -54 => 158 (=222-54).
=> 1588
1588 - 1110 = 478
478 - 444 = 34.

Done.
Basically, in reach step I ask myself what factor do I need to subtract to get rid of the first (= leftmost) digit.
If the result is negative, I add 222 to make it positive.

In the last step for example - 1558 - I ask what factor will make sure I get rid of the ‘1’.
That is why I subtract 1110 first (instead of looking for the biggest factor below 1558).
This is not the fastest way.
It just makes it easier for me to focus on getting rid of the leftmost digit

Exactly for that reason do I take the extra steps.

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I know. It’s just that @benjamin1990 asked how exactly to do this calculations, hence I wrote down step by step here how I would have done it. Actually this example alone would probably have taken me ten to twenty minutes (or more) when done in mind only. I really admire mental math competitors who can do such tasks in no time.

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In the western world, we are at disadvantage since we do not use abacus or soroban.

Abacus/Soroban/Anzan would make such divisions extremely easier.

To compensate for that lack of abacus, we western competitors try to be more creative and memorize more things than the Asians.

For example, I know by heart all factor products up to 1000,
(23 x 43 = 989 etc.) and all the squares and cubes up to 100. (87 cubed
658503 etc).

It is all about effort, creativity, positive attitude to numbers and willingness to commit such info to memory, and learning new tricks.

No one is born with super speed in mental math.

Speed in mental math is a side-effect after doing many drills for many years/ (even decades). One has to train to learn the methods, and speed will come naturally, afterwards.

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That is not true.

It took me around 1-2 minutes which I think is pretty decent for the first time. I also calculated 889014113748268 mod 7 in my head just to see if smaller numbers really are as easy as Nodas mentioned and it is, this one took me around 20 seconds.

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I found myself actually taking a different approach.

Half of 222 is 111, which has a pretty neat table for this.

111, 222, 333, 444, etc. It requires no thinking and the pattern is only broken at 1110, which I wont use in the division.

After that is it one of two possibilities, which only come to play at the end, because everything divisible by 222 is also divisible by 111.

You either get an even amount from the division, which you can just half if someone asks for it. No changes to the remainder.

You can also get an uneven amount at the end, which means you remove one, and then half it. The remainder gets 111 added.

The final removal is 444, or 4x111, leaving 34. 444 being even means nothing changes.

This way it took me around 40 seconds, so someone with more experience will probably be able to do it faster.

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That is not true.

Of course it is true. There has never been a baby in the history of mankind that can calculate out of womb.

The great British philosopher Locke talked about the mind being a blank slate about 400 years, and this is extremely true.

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You’re right @Nodas but there are people out there that do mental calculation in a way we can call it innate, people who are so fast…

I agree that everyone is born with a blank slate; a blank slate of knowledge.

Yes, we aren’t born with the understanding of concepts like multiplication, division etc but when we teach young children the steps of arithmetic, a clear pattern emerges. In an equal playfield, no abacus, no techniques or shortcuts, (in my case even a language barrier), there are some children who are faster than others at mental math.

Just like some children are more creative than others, or smarter, or nicer or more athletic, etc.

When I was a kid in elementary school, my math teachers were always telling me to workout my problems, explain the steps that I took to solve an arithmetic problem because all I would write down were the answers. At the time I always felt annoyed by them because I had to write everything twice. I often told them that I did the steps they wanted me to write down and that I didn’t need to write it out, I didn’t want to write them out. Even after writing down the steps I took, they would still tell me to write all the steps down. Years later, I realized why they kept telling me to write everything down. It wasn’t to check if I did it right, it was because they couldn’t understand how I could do these steps without writing anything down and so fast.

Today I know that children like that are very, very rare. Adults even rarer. I used to have a lot of doubts about my abilities, sometimes I still do but they keep proving themselves to be real and I’ve had them as long as I can remember. If you don’t call that born-abilities then I must have been upgraded in a lab as a 2 year old or something.

If I wasn’t me, I would totally agree with you because I haven’t met anyone in real life like this either. I have found some savants online though, like the chinese rainman and a few others. However they only confirm how almost non-existent and rare the phenomenon is.

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Totally agree with you.

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Well, I agree that humans are different, but problem is that environment also has great impacts on who we become today, and I think it’s hard to tell whether environment or intelligence dominated.
In your case you don’t like to write answers of arithmetic down, and you indeed did it, that already differs from most of the student, they still choose to write it down and their ability doing mental math might never progress. And possibly they would never have motivation to do it later. So it’s not ‘fair’ when game started.
And later it might become a illusion that only you could do this stuff, others just stood there like goldfishes. (just probably, maybe you’re indeed smart when you was born, but we lack the means to verify it.) The differences between you and people you interact with will make you find out more differences, and they will convince you even more. And what happened next is predictable. But whatever, this is just a theory.
I think you should try to practice your ability in the future. Maintaining its ‘originally’ is unfair for others , also unfair for yourself. If one day you could do better than the top calculators(If you thrive in mental arithmetic, of course you can), then I think no one could doubt what you have claimed

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I see what you mean and you have a point but in my case things were a bit different. The teachers eventually gave me math material that was a few classes ahead of my peers. Also, I wasn’t born here. Most of my years at elementary school I struggled with the dutch language because when I moved here I could only speak portuguese but perhaps autism might have also played a role. However, even with my struggles with the dutch language, I managed to be 1 of the 2 people in my class who got a VWO advice from the Citotoets. VWO is the highest form of education you can take at middle/high school here in the Netherlands and the Citotoets is a test that children make in the last year of elementary school to see which educational route is the best for them. I remember asking my mother’s boyfriend at the time if I could go to VWO because my best friend also went to VWO. For the longest time I didn’t know what VWO even was lol, nobody ever explained it. He told me that not only would I go to VWO, the school said that I HAD to go to VWO. Of course I was very happy because I could still see my best friend.

There were a lot of factors in my life that normally would hold a person back but, despite of my autism, language barrier and troubled youth I still came out on top. My psychologist told me last year that I was very lucky to have ended as such a decent person too because normally people who had a similar youth experience would’ve ended way worse. I agreed because at one point in my life, I almost ended ‘way worse’.

It’s a dilemma because if I train my abilities, people might think my abilities are due to training but they aren’t. Of course I could tell people that it’s innate but I like to rule out everything. But I am slowly changing my mind because I would like to see my abilities’ full potential too. Imagine if I could compete at the top without any techniques? That sentence alone feels controversial, lol.

Nice and clear explanation.

It’s indeed true that VMO is dominated by pupils with better-educated parents a non-migrant background, and you also make it, which is remarkable, but if I remember correctly, there are almost 20% of students in the Netherlands graduated with a VWO qualification every year. Although students with this qualification is pretty decent, most of them are not geniuses by any means.

In my opinion, you care too much about what others think, but sometimes people don’t think like that in fact.

Sorry I didn’t find anything controversial in this sentence, perhaps it’s due to my poor English, but if I understand correctly, you mean “without techniques, it’s impossible to compete at the top”.
I don’t think so. Take mental arithmetic for example, I think it’s definitely possible to achieve.

Just do it. It doesn’t really matter whether someone else is using techniques or not. If you just stay at where you are, at last you could only convinced yourself that you have the great ability.

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Well, it does seem impossible. Let me give an example. My fastest time for a 4 by 4 digit multiplication is 15 seconds, like I mentioned before I don’t use any shortcuts method or anything like that, I do it the way you are taught in elementary school. If you go out on the streets and ask 10,000 people to calculate 3487 x 6359 in their head in less than 60 seconds, you won’t get an answer. If you do get an answer, the chances are high that that person is using a method like the criss-cross method or you have just found another rarity.

So the average person, who can barely hold the 8 digits of the multiplication in their head, can’t even do such a calculation.

Now realize that in the top of mental math, the best can solve a 4 by 4 in less than 5 seconds, some maybe in 3 seconds.

That is 5 times faster than I am. If I try very hard I might get to 10 seconds but 3? I can only dream of.

Which is awesome cuz you guys make me feel normal, lol. It’s humbling and I feel connected.

I am not the only one who is convinced. My teachers, family, classmates, work colleagues, psychologists, almost anyone who witnessed me in action is convinced. Just a few weeks ago, I was told by my boss how he wished he had my brain. He said I was like a walking and talking computer. His children also have autism, perhaps that’s a reason why he is so impressed.

Anyway that is not what this conversation is about.
There are people who are born with an innate ability for speed mental math. Perhaps not many but they do exist.

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Who is this top mental calculator that can solve in 3 seconds ?

I think Rudiger Gamm could do 4 by 4 multliplications in 3 seconds or 5, I don’t recall exactly. It was mentioned by someone who interviewed him. It took him 15 seconds to solve an 8 x 8 multiplication. He didn’t need to think about 2 x 2, almost the same for 3 x 3. He’s a monster.

Rudiger Gamm might also be someone with a special brain. He can speak backwards and his ability to calculate two digit numbers to the power of 200 goes beyond just memorization. He said he does calculate them and that it could take him 10 minutes to speak the answer of, for example, 87^200.

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