Question
Six professors begin courses on Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday, respectively, and announce their intentions of lecturing at intervals of 3, 2, 5, 6, 1, and 4 days, respectively. The regulations of the university forbid Sunday lectures (so that a Sunday lecture must be omitted). When first will all six professors find themselves compelled to omit a lecture? Hint: Use the CRT.
Answer
To determine when all six professors will find themselves compelled to omit a lecture, we need to find the least common multiple (LCM) of the intervals at which they lecture.
The intervals are: 3, 2, 5, 6, 1, and 4 days.
To find the LCM, we can list the multiples of each interval until we find a common multiple.
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, ...
Multiples of 6: 6, 12, 18, 24, 30, 36, ...
Multiples of 1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, ...
It can be observe that the first common multiple among all intervals is 12.
Therefore, all six professors will find themselves compelled to omit a lecture after 12 days.