General Han would like to count the exact number of soldiers.

Question

He orders his soldiers to form groups of 9, then groups of 11, and then groups of 17. The remainders are 2, 1, and 10 respectively. He knows there are 5000 to 6000 soldiers. How many soldiers does he have?

Answer 1

Let $x$ be the number of sodiers. One has $$x \equiv 2 \mbox{ (mod $9$)},$$ $$x \equiv 1 \mbox{ (mod $11$)},$$ $$x \equiv 10 \mbox{ (mod $17$)}.$$

From the Chinese Remainder Theorem, one has $$x \equiv 452 \mbox{ (mod $1683$)}.$$

Because $5000 \leq x \leq 6000$, then $x = 5501$.