Reading Worksheets for 2nd Graders Worksheets - Free Printable

It looks like you've uploaded a text-based image of a riddle or logic puzzle titled "Rameses' Train". Let's solve it step by step.

---

🔍 The Puzzle:

> Rameses and Elise were having fun on the swings at the park as a kid named Joe. They both loved the feeling of up and down, but when they would swing high up over the ground, Elise’s little sister dropped them separately in the dirt to stop their swinging.
>
> "Now that gives the latest answer to get me out of this mess," said Rameses.
>
> "What are you getting?" asked Elise.
>
> "I need an example problem," replied Rameses. "That's my favorite kind."
>
> "Let's see... what's the next number?"
>
> The two girls brought the paper and then started to notice: There was a sequence where the numbers went from 1 to 5, then to 6, then to 7, then to 8, then to 9, then to 10, then to 11, then to 12, then to 13, then to 14, then to 15, then to 16, then to 17, then to 18, then to 19, then to 20, then to 21, then to 22, then to 23, then to 24, then to 25, then to 26, then to 27, then to 28, then to 29, then to 30, then to 31, then to 32, then to 33, then to 34, then to 35, then to 36, then to 37, then to 38, then to 39, then to 40, then to 41, then to 42, then to 43, then to 44, then to 45, then to 46, then to 47, then to 48, then to 49, then to 50, then to 51, then to 52, then to 53, then to 54, then to 55, then to 56, then to 57, then to 58, then to 59, then to 60, then to 61, then to 62, then to 63, then to 64, then to 65, then to 66, then to 67, then to 68, then to 69, then to 70, then to 71, then to 72, then to 73, then to 74, then to 75, then to 76, then to 77, then to 78, then to 79, then to 80, then to 81, then to 82, then to 83, then to 84, then to 85, then to 86, then to 87, then to 88, then to 89, then to 90, then to 91, then to 92, then to 93, then to 94, then to 95, then to 96, then to 97, then to 98, then to 99, then to 100.
>
> Then came the question:
>
> "This sequence is increasing by one, just about every time. But now, how many numbers are there between 1 and 100 that are not divisible by 2, 3, or 5?"

---

✔ Goal:

Find how many integers from 1 to 100 are not divisible by 2, 3, or 5.

This is a classic inclusion-exclusion principle problem.

---

🧮 Step-by-step Solution:

We want to count the numbers from 1 to 100 that are not divisible by 2, 3, or 5.

Let’s define:
- Total numbers = 100
- Let A = numbers divisible by 2
- Let B = numbers divisible by 3
- Let C = numbers divisible by 5

We want:
Total – (A ∪ B ∪ C)
That is:
100 – |A ∪ B ∪ C|

Using inclusion-exclusion:
> |A ∪ B ∪ C| = |A| + |B| + |C| – |A ∩ B| – |A ∩ C| – |B ∩ C| + |A ∩ B ∩ C|

---

#### 🔹 Count each:

1. |A|: divisible by 2 → ⌊100/2⌋ = 50
2. |B|: divisible by 3 → ⌊100/3⌋ = 33
3. |C|: divisible by 5 → ⌊100/5⌋ = 20

4. |A ∩ B|: divisible by LCM(2,3)=6 → ⌊100/6⌋ = 16
5. |A ∩ C|: divisible by LCM(2,5)=10 → ⌊100/10⌋ = 10
6. |B ∩ C|: divisible by LCM(3,5)=15 → ⌊100/15⌋ = 6

7. |A ∩ B ∩ C|: divisible by LCM(2,3,5)=30 → ⌊100/30⌋ = 3

---

#### 🔹 Plug into formula:

|A ∪ B ∪ C| = 50 + 33 + 20 – 16 – 10 – 6 + 3
= 103 – 32 + 3 = 74

So, 74 numbers between 1 and 100 are divisible by 2, 3, or 5.

Therefore, numbers not divisible by 2, 3, or 5 =
100 – 74 = 26

---

✔ Final Answer: 26

---

📌 Explanation:

There are 26 numbers between 1 and 100 that are not divisible by 2, 3, or 5.

These are numbers that are coprime to 30, since 2×3×5=30, and such numbers are relatively prime to 30.

Examples include: 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97.

Note: Some like 49 = 7², 77 = 7×11, etc., are not divisible by 2,3,5 even though composite.

---

💡 Summary:

> Answer: 26

This is the number of integers from 1 to 100 that are not divisible by 2, 3, or 5.

Let me know if you'd like a list of those 26 numbers!