There are between 500 and 600 soldiers in a battalion. We know when they are divided into groups of six, eight, and 11 that there are three left over in each case. How many soldiers are there in the battalion?
Let $n$ be the number of soldiers. We are given
$$ k[6,8,11] = n - 3 $$
$$ 500 < n-3 < 600 $$
We note that $[6, 8, 11] = 264$, and so $k=2$ gives $k[6,8,11]=528=n-3$.
Which means $n=531$, the number of soldier sin a battalion.