# Remainder when dividing by 990: Chinese Remainder Theorem

What is the remainder when the number

$101102103104105...996997998$

is divided by 990?

(The digits of the number are just the digits of all the integers from 101 to 998 inclusive, written side-by-side.)

I think I would have to use chinese remainder theorem and factor 990.

990= $2$$\cdot$$3^2$$\cdot$$5$$\cdot$$11$

Do I just create a system of 4 modular equations with different modulos as the factors listed above?

• I wouldn't bother with $2$ and $5$, instead doing $10$, and doing that just by writing down the answer. – Angina Seng May 16 '17 at 6:23

First, note that $1000\cong 10\pmod{990}$. Then, $$1000^2\cong 100\pmod{990}$$ so $$1000^3\cong 1000\cong 10\pmod{990}$$ We can show by an easy induction that $1000^k\cong 10\pmod{990}$ if $k$ is odd, and $1000^k\cong 100\pmod{990}$ if $k$ is even (both for $k\geq 1$). Then, the number in question, which we can rewrite as $$\sum_{k=0}^{897} (998-k)1000^k = 998+\sum_{i=0}^{448} (997-2k)1000^{2i+1}+\sum_{j=0}^{447} (996-2k)1000^{2j+2}$$ is congruent to $$998+\sum_{i=0}^{448} 10(997-2k)+\sum_{j=0}^{447} 100(996-2k) \\ = 998+10(448+1)(997-448)+100(447+1)(996-447) \\ = 998+10\cdot 449\cdot 549+100\cdot 448\cdot 549$$ We can calculate the remaining modular division by hand to find that this is congruent to $548\pmod{990}$.
• Note that dividing the number mod $11$ or mod $99$ is about as complex as dividing it mod $990$, since you are still left with sums that look like the ones above. Therefore, I believe that the Chinese remainder theorem does not make this problem simpler. – Michael L. May 16 '17 at 7:26
• Can we use the CRT? If we let this number=x then we get three congruences. $x \equiv k mod 10$, $x \equiv k mod 9$, $x \equiv k mod 11$. We then solve for k – shrindle May 17 '17 at 4:23
• Sure, but calculating $k$ divided by $11$ is by itself almost as complicated as calculating it divided by $990$ (in consideration of the fact that we will have a sum similar to the above). I don't see CRT as making this problem simpler. – Michael L. May 19 '17 at 13:36