What counts as a broken clock anyway? If it just stops then yes, that person is right, of course - but! I have a wristwatch that is always out of time. Fix it and leave it for a few days? It's going to show some unrelated time again.
Assume the clock moves v times as fast (slow) as "real time" and starts right. Will it still be right? The start doesn't count.
t = vt (mod 12)
|v-1|t=0 (mod12)
t=12n/|v-1| where n is a whole number between 0 and infinity
0<t<=12 but n can really be as big as we want theoretically. When n is big enough, which it will be eventually, the last equation will be wrong