2012
Problem - 1395
Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?
Answer
C
While MOD can only be applied to integer, but conceptually this problem is equivalent to solving $$4.4t\equiv 4.8t\equiv 5.0t \pmod{500}$$
Subtracting $4.4t$ leads to $$0\equiv 0.4t\equiv 0.6t \pmod{500}$$
This means that we need to find a $t$ such that $0.4t$ and $0.6t$ are both multiples of $500$. The first few $t$s which satisfy this condition are $1250$, $2500$, $\cdots$. We find $\boxed{2500}$ is a solution.