Question

Let N = 123123123... (300 digits).
Find the remainder when N is divided by 504

Answer 1

\(\sum_{n=0}^{99}123\times 10^{3n} \mod 504\)

Answer 2

And
504= 3*2^3*23

Answer 3

504 = 7*8*9

Answer 4

Ok so we need find construction of 10^3n mod 504

Answer 5

Lol that was stupid xD

Answer 6

Ok there is a pattern
123+ 123(-8+64-8+64+...)

Answer 7

123+123(49*56)+123*-8. Mod 504

Answer 8

So we would have
123(1+2744-8) mod 504
123(1+224-8) mod 504
123*217 mod 504

Answer 9

I would say remainder is 483
Hope it's right calculating mentality frustrated me xD

Answer 10

I would like also to see fancy theorems or algorithms it would be fun

Answer 11

483 is correct!
im actually looking for a more elementary method without using so much of number theory as this particular question was asked in one of the highschool level competitions..

Answer 12

I was looking for a way to combine the fact that \[504=7*8*9=n*(n+1)(n+2)=1n^3+3n^2+2n \text{ for n=7}\] into something that exploits that 1,3,2 to somehow do magical things to the 1,2,3 while avoiding touching the tempting geometric series.

Answer 13

Or maybe it's just a coincidence... haha... I sort of took that 504=7*8*9 to be almost a type of hint.

Answer 14

Ahh that might give some easy criterian for divisibility test for 7, not so sure, still thinking..

Answer 15

I think I got it.

Answer 16

thats a neat approach! let me see if i understand it correctly..

for n=5, im getting remainder =291 ?
http://www.wolframalpha.com/input/?i=123*10%5E%283*%285-1%29%29+%2B+123123123123+mod+504

Answer 17

Oooh, oops. I hate my calculator.

Answer 18

this is easy with chinese remainder theorem
i just don't understand why they give these problems in highschool level competitions

Answer 19

Sorry, I got everything wrong. That can never be zero.

Answer 20

Which competition?

Answer 21

this question is everywhere. i saw this for the first time on brainly last night, google gave me million+ results.. mostly from various competitions in india

Answer 22

but i couldn't find any satisficatory solution anywhere online

Answer 23

lol, brainly? seriously? D:

Answer 24

Haha thats the reason i need to dumb it down to highschool level, otherwise they are deleting my answers

Answer 25

That's foolish. If a method provides a legitimate solution, no matter how incomprehensible it is, it should be left there. Sounds like a dumb reason to delete answers.

Answer 26

Watching the hundred feet journey, I have some other methods though

Answer 27

`Reason Hi! I'm sorry but your answer lacks some vital details and therefore, it is being deleted. Please elaborate your answer further and repost it.`

Answer 28

please post after the movie ikram

Answer 29

i thought it was that easy thats why i used less NT hmmm other method is deleling with it as series

Answer 30

hmm no not interesting anymore.. i hate hight school anyway

Answer 31

dealing*

Answer 32

hmm

Answer 33

hey are you from brainly

Answer 34

yeah..

Answer 35

welcome! :)

Answer 36

thnks

Answer 37

what is brainly ?

Answer 38

seems i have account there :O
since when XD

Answer 39

http://brainly.com/profile/ikram002p-98608

Answer 40

yes that site i think it forces you to create account to see answer to a problem

Answer 41

I see