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Let N be the integer of putting all natural numbers from 1 to 35 together

Therefore, N = 1234567891011121314...35

Find the remainder when N is divided by 36

Here's what I have done:

$$N\equiv1(\textrm{mod}\ 2)$$ $$N\equiv0(\textrm{mod}\ 3)$$ $$N\equiv-1(\textrm{mod}\ 4)$$ $$N\equiv0(\textrm{mod}\ 9)$$

We are finding $N (\textrm{mod}\ 36)$, so how do we put all those congruences up there together and find the answer?

Cyh1368
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    Since $36=4\times9$ and $\gcd(4,9)=1$, you can use the Chinese remainder theorem to convert the congruences modulo $4$ and $9$ into a single congruence modulo $36$. – Greg Martin Oct 30 '21 at 06:40
  • @GregMartin I have read the wikipedia page you linked, but I still do not understand how to combine mod4 and mod9 into mod36. If possible, can you provide a detailed solution? Thanks. – Cyh1368 Oct 30 '21 at 06:48
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    The link provided by the comment of @GregMartin does explore solving simultaneous congruency problems. However, this exploration is somewhat complicated. You may wish to consider finding the right elementary Number Theory book, opening it to page 1, and proceeding from there. You would certainly want to check first that the book explores the use of the Chinese Remainder Theorem to construct a solution. However, I suspect that the Chinese Remainder Theorem is a very common topic in elementary Number Theory. Personally, I went with "Elementary Number Theory", Uspensky and Heaslett, 1938. – user2661923 Oct 30 '21 at 08:08
  • @Cyh1368 that's just an algorithm you need to follow to solve (more complicated) congruences. It's no rocket science. Just google "Chinese Remainder Theorem" and you'll find lots on it. Look at a few examples, practise a few problems, and soon you'll get what's going on. Alternately, as user2661923 says, pick a book and follow it from page 1. I personally went for the one by David Burton. – Sayan Dutta Oct 30 '21 at 09:17
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    I know the basic principles of the Chinese Remainder Theorm, as well as solving basic multi-congruence problems, but I do not know that it can be used to solve more complicated problems as shown on the wikipedia page. However, thanks for all of you's advice. – Cyh1368 Oct 30 '21 at 13:19

1 Answers1

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You got, $N\equiv0(\text{mod } 9)$. So, $N$ is divisible by $9$. Also, you got $N\equiv-1(\text{mod } 4)$. But, since $9$ and $4$ do not share any common factors, you must have $\frac N9\equiv-1(\text{mod } 4)$. So, $$\frac N9=4m+3$$ which gives $$N= 36m+27$$ and hence $$N\equiv 27(\text{mod }36)$$ Is that fine?

Sayan Dutta
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