Sheep left after three raids

Three gangs of rustlers raid a flock in turn. The first takes a third of the flock plus a third of a sheep. The second takes a quarter of what is left plus a quarter of a sheep. The third takes a fifth of the new remainder plus three-fifths of a sheep. Afterwards $409$ sheep remain. How large was the original flock?

1025. Work backwards. Before the third raid, $\tfrac45n_2-\tfrac35=409$ gives $n_2=512$; before the second, $\tfrac34n_1-\tfrac14=512$ gives $n_1=683$; before the first, $\tfrac23n-\tfrac13=683$ gives $n=1025$. Each raid removes a whole number of sheep ($342$, $171$, $103$).