What is the remainder when 10,987,654 is divided by 2,103?

Question

What is a simple way to solve this problem? I solve this problem by actually dividing $10,987,654$ by $2,103$, which should not be a simple way.

What is the remainder when $10,987,654$ is divided by $2,103$?

Answer 1

As is tagged in your question, you can use the Chinese Remainder Theorem. We have that

$$2013=3\cdot11\cdot61$$

So we wish to find $$a\mod k$$ for $a=10987654$ and each of $k\in\{3,11,61\}$. These are fairly easy to calculate, and we get the equivalences

$$a\equiv1\mod 3$$

$$a\equiv7\mod 11$$

$$a\equiv29\mod 61$$

We now note that the number $29$ solves each of the second and third congruences, so we just want to add multiples of $671=11\cdot61$ to it to get it to be $\equiv1\mod3$. We have $$29+671=700\equiv1\mod3$$

so we get that

$$10987654\equiv 700\mod 2013$$

Answer 2

$\pmod {2103} :$

$\color{red}{10 987} 654 \equiv \color{red}{472}654$

$\color{red}{4726}54\equiv \color{red}{520}54$

$\color{red}{5205}4\equiv \color{red}{999}4$

$\color{red}{9994}\equiv \color{red}{1582}$

So yeah it's boring but not too hard without a calculator.