'6 professors begin courses of lectures on Monday, Tuesday, Wednesday, Thursday, Friday and Saturday, and announce their intentions of lecturing at intervals of 2,3,4,1,6,5 days respectively. The regulations of the University forbid Sunday lectures (so that Sunday lecture must be omitted). When first will all six professors find themselves compelled to omit a lecture?'
This is the question I'm struggling with, I believe it can be solved using the Chinese Remainder theorem, but am really going no where with it. Any progress I make I will post, but I have not yet done anything worth adding to this. Any help would be greatly appreciated!
EDIT: Ok, so I have reduced the problem down to finding the minimum positive value such that, d=11(mod 12), 6(mod 5), 0(mod 7).
Best Answer
Luckily seven is a prime, so all the professors will need to skip every seventh lecture. Let the coming Sunday be number zero, let $i$ be the Sunday $i$ weeks into the future, and let $N$ denote the number of a Sunday, on which they all need to skip a class. Prof. Monday will need to skip this Sunday and every second Sunday thereafter, so $N\equiv 0 \pmod 2$. Prof. Tuesday would first like to preach on Sunday $\#1$ exactly $12=4\cdot3$ days after his course commences, so $N\equiv 1\pmod 3$. Prof. Wednesday will miss an opportunity on the coming Sunday as well, so $N\equiv 0\pmod 4$. The eager beaver prof. Thursday will be irritated on every single Sunday, and can thus be left out of the reckoning. The lazy professor Friday will first be annoyed (or relieved?) on Sunday $\#4$ (four full weeks plus two days from Fri to Sun = $30=5\cdot 6$ days), so $N=4\pmod 6$. Professor Saturday's first miss is $15=3\cdot5$ days after his course has started on Sunday $\#2$, so $N\equiv 2\pmod 5.$
BUT I'm not going to solve this for you. Check the above, and then use the CRT to find the possible values of $N$. For extra credit you should observe that it was not a priori clear that the six professors would all miss a lecture on the same Sunday. A number of miraculous fits occurred, so I give some due credit to whoever composed this problem.