I am studying a book right now, and I'm having a difficulty understanding a (solved) problem regarding congruent modulo. Below I will list the problem and what I have understood of the problem, along with what it is I don't understand:
Problem: Suppose you watch an entire season of your favorite television show in one sitting, starting at midnight. There are 25 episodes, each lasting 3 hours. At what time of the day are you done? Answer: the hour of completion is (25 x 3) mod 24, which (since 25 ≡ 1 mod 24) is 1 x 3 mod 24, or three o'clock in the morning.
My own understanding: I understand that a number x congruent to another y modulo z, means that x and y are in the same equivalence class when they are modulo z, or put on other words that x - y is divisible by some multiple of z.
Difficulty of understanding: Although I believe I have got a grasp on the congruent modulo, I don't understand how the author concludes that
the hour of completion is 1 x 3 mod 24.
Can someone explain to me what I appear to be missing?
The congruence relation respects addition and multiplication. This means that if $a\equiv x$ (mod m) and $b \equiv y$ (mod m), then $a+b \equiv x+y$ (mod m) and $ab \equiv xy$ (mod m). Since $25 \equiv 1$ (mod 24), we have $25*3 \equiv 1*3 \equiv 3$ (mod 24).