TD - Black-Scholes equation, spot price and option price

Question
Show that the Black-Scholes equation remains invariant under the scaling
S
0 = αS where α > 0 is a constant.
A put option with strike K is written on an asset which pays out on a single,
discrete yield q at time td < T, where T is the expiry date of the put.
Explain why the spot price jumps from S to (1 − q)S as the dividend date
is crossed, but the option price remains continuous. Denote the option price
by P(S,t; K, T).
Let PBS(S,t; K, T) denote the usual Black-Scholes value for a put option on
an asset which pays no dividends and has strike K, expiry T. Show that
P(S,t; K, T) =
(
PBS(S,t; K, T) if td < t < T,
(1 − q)PBS(S,t; K/(1 − q), T) if 0 ≥ t < td.