One chicken lays an egg every 2 days, another every 3 days and a third chicken lays an egg every 5 days. They all lay an egg today. How many days in the next 101 days will EXACTLY 2 chickens be laying an egg?

It sounds as if you need to find a good calendar with big blocks and three pens of different colors.

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First, you have to find the least common multiple of the number of days between eggs (2,3,4) to get 30. Then, you make a list of numbers 1-30:

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

To solve this problem, we can use the concept of finding the least common multiple (LCM) of the numbers 2, 3, and 5. The LCM will give us the number of days it takes for all three chickens to simultaneously lay eggs again.

First, let's find the LCM of 2, 3, and 5:
- The prime factors of 2 are 2.
- The prime factors of 3 are 3.
- The prime factors of 5 are 5.

To find the LCM, we take the highest exponent of each prime factor:
2^1 × 3^1 × 5^1 = 2 × 3 × 5 = 30.

So, it takes 30 days for all three chickens to simultaneously lay eggs.

Now, let's divide 101 days by the 30-day cycle to determine how many cycles occur in that period:
101 ÷ 30 = 3 remainder 11.

This means that there are 3 complete 30-day cycles, with an additional 11 days remaining.

Next, we need to find out how many days within the 11 remaining days exactly 2 chickens lay eggs.

Out of the three chickens, every two-day cycle, two-chicken combinations lay eggs on the following days:
- Chicken 1 and Chicken 2: Days 2, 4, 8, 10, 14, 16, 20, 22, 26, 28.
- Chicken 1 and Chicken 3: Days 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
- Chicken 2 and Chicken 3: Days 5, 10, 15, 20, 25, 30.

If we analyze the pattern, we see that the 2-chicken combinations lay eggs every 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, and 30 days.

Within the remaining 11 days, the combinations 2 and 4 (Chicken 1 and Chicken 2) and 3 and 6 (Chicken 1 and Chicken 3) will occur. The 10th day is the next time both combinations occur.

Therefore, within the next 101 days, there will be exactly 2 chickens laying eggs on 10 days.