Question

A company employed three security men. A, B, C, for rotational duties. A starts his duty in the first week of the year, followed by B and the C. After how many weeks will they repeat their duties from the first week of a month?

Answer 1

They will repeat their duties from the first week of a month after 7 weeks. A starts in the first week, followed by B and then C. The rotation completes in 3 weeks. For the cycle to start over at the first week of a month, it requires an additional 4 weeks, totaling 7 weeks. So, every 7 weeks, they will return to their initial duty rotation.

Explanation:

A, B, and C follow a 3-week duty rotation, and after three weeks, they would have completed their first cycle. To start from the first week of a month again, you need an additional 4 weeks, making it a total of 7 weeks for the entire cycle to repeat.

Answer 2

To determine when the three security men will repeat their duties from the first week of a month, we need to find the least common multiple (LCM) of the number of weeks in a year and the number of security men.
There are 52 weeks in a year, and the three security men take turns in a rotational pattern. Let's represent this pattern as ABC.
To find the LCM, we need to list the multiples of 52 until we find a multiple that is also a multiple of 3. The multiples of 52 are:
52, 104, 156, 208, 260, 312, 364, 416,
468, 520, 572, ...
Now let's check which of these multiples are also multiples of 3.
52 is not a multiple of 3.
104 is not a multiple of 3.
156 is a multiple of 3.
So, after 156 weeks, A, B, and C will repeat their duties from the first week of a month.
To illustrate this, let's look at the first 156 weeks:
all ~ 80
X
Week 1: A
Week 2: B
Week 3: C
Week 4: A
Week 5: B
Week 154: C
Week 155: A
Week 156: B
Therefore, after 156 weeks, the pattern ABC will repeat, and they will start their duties from the first week of a month