Answer:
B. 19 min 15 sec
B. 13 min 25 sec
Step-by-step explanation:
Door 1 opens every 1 min 45 sec, or 105 sec.
Door 2 opens every 1 min 10 sec, or 70 sec.
Door 3 opens every 2 min 55 sec, or 175 sec.
Door 4 opens every 2 min 20 sec, or 140 sec.
Door 5 opens every 35 sec.
The greatest common factor is 35 seconds, so we can measure the time in units of 35 seconds.
Door 1 opens every 3 units.
Door 2 opens every 2 units.
Door 3 opens every 5 units.
Door 4 opens every 4 units.
Door 5 opens every 1 unit.
The least common multiple of 3, 2, 5, 4, and 1, is 60. So every 60 units, all five doors will open, and the guard will look down the corridor to check on the prisoner. Douglas must escape before this time.
In order to escape in the shortest time possible, Douglas should time his escape so that each door opens 1 unit after the door before it. It takes Douglas 20 seconds to move from one door to another, so he will have enough time to get to the next door before it opens.
Let's say Douglas starts moving when Door 1 opens for the nth time. In other words, 3n units have passed before he starts moving. That means Door 2 should open after 3n + 1 units. Door 3 should open after 3n + 2 units. Door 4 should open after 3n + 3 units. And Door 5 should open after 3n + 4 units.
Since Door 2 opens every 2 units, 3n + 1 should be a multiple of 2.
Since Door 3 opens every 5 units, 3n + 2 should be a multiple of 5.
Since Door 4 opens every 4 units, 3n + 3 should be a multiple of 4.
Since Door 5 opens every 1 unit, 3n + 4 should be a multiple of 1.
By trial and error, n = 11.
So Douglas starts moving after 33 units, or 1155 seconds, or 19 min 15 sec.
Douglas clears the fifth door after 37 units, which leaves 23 units to spare, or 805 seconds, or 13 min 25 sec.
