##### Question 7.9.8

I have some marbles which I’m trying to arrange in a rectangle, but so far I’ve had no luck.

• If I arrange them in rows of $4$, there are $3$ marbles left over.

• If I arrange them in rows of $5$, there is $1$ marble left over.

• If I arrange them in rows of $7$, there are $6$ marbles left over.

I estimate that there are a bit more than $200$ marbles. How can I arrange all of them in a rectangle? (I won’t accept the trivial arrangement of rows of $1$ marble.)

If you prefer a more abstract statement of the problem, here it is: find the smallest $n\gt 200$ such that \begin{align*} n &\equiv 3 \pmod{4}\\ n &\equiv 1 \pmod{5}\\ n &\equiv 6 \pmod{7}. \end{align*} What are the divisors of $n$?

Similar problems were mentioned in The Mathematical Classic of Sunzi from fifth century China. There is a theorem about questions like this, known as the Chinese remainder theorem. Show hint