How many numbers between 50 and 1290 are not divisible by 2 and 3 and 5

To find how many numbers between 50 and 1290 are not divisible by 2, 3, or 5, we can use the **principle of inclusion and exclusion (PIE)**.

### Step 1: Determine the Total Count of Numbers

The total number of integers between 50 and 1290 can be calculated as:
\[
\text{Total numbers} = 1290 - 50 + 1 = 1241
\]

### Step 2: Count Multiples of 2, 3, and 5

Let:
- \( A \) be the set of multiples of 2.
- \( B \) be the set of multiples of 3.
- \( C \) be the set of multiples of 5.

We want to find the size of the set \( A \cup B \cup C \), which represents numbers divisible by 2, 3, or 5. We will use the formula:
\[
|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|
\]

### Step 3: Use Arithmetic Sequences to Find Each Set's Size

1. **Multiples of 2 (\( A \))**:
   The smallest multiple of 2 within the range is 50, and the largest is 1290. The sequence is:
   \[
   50, 52, 54, \ldots, 1290
   \]
   This is an arithmetic sequence with:
   - First term \( a = 50 \)
   - Common difference \( d = 2 \)

   The number of terms, \( n \), can be found by:
   \[
   n = \frac{1290 - 50}{2} + 1 = 621
   \]
   So, \( |A| = 621 \).

2. **Multiples of 3 (\( B \))**:
   The smallest multiple of 3 within the range is 51, and the largest is 1290. The sequence is:
   \[
   51, 54, 57, \ldots, 1290
   \]
   \[
   n = \frac{1290 - 51}{3} + 1 = 414
   \]
   So, \( |B| = 414 \).

3. **Multiples of 5 (\( C \))**:
   The smallest multiple of 5 within the range is 50, and the largest is 1290. The sequence is:
   \[
   50, 55, 60, \ldots, 1290
   \]
   \[
   n = \frac{1290 - 50}{5} + 1 = 248
   \]
   So, \( |C| = 248 \).

4. **Multiples of 6 (\( A \cap B \))**:
   These are multiples of both 2 and 3, which means multiples of 6. The sequence is:
   \[
   54, 60, 66, \ldots, 1290
   \]
   \[
   n = \frac{1290 - 54}{6} + 1 = 206
   \]
   So, \( |A \cap B| = 206 \).

5. **Multiples of 10 (\( A \cap C \))**:
   These are multiples of both 2 and 5, which means multiples of 10. The sequence is:
   \[
   50, 60, 70, \ldots, 1290
   \]
   \[
   n = \frac{1290 - 50}{10} + 1 = 125
   \]
   So, \( |A \cap C| = 125 \).

6. **Multiples of 15 (\( B \cap C \))**:
   These are multiples of both 3 and 5, which means multiples of 15. The sequence is:
   \[
   60, 75, 90, \ldots, 1290
   \]
   \[
   n = \frac{1290 - 60}{15} + 1 = 82
   \]
   So, \( |B \cap C| = 82 \).

7. **Multiples of 30 (\( A \cap B \cap C \))**:
   These are multiples of 2, 3, and 5, which means multiples of 30. The sequence is:
   \[
   60, 90, 120, \ldots, 1290
   \]
   \[
   n = \frac{1290 - 60}{30} + 1 = 41
   \]
   So, \( |A \cap B \cap C| = 41 \).

### Step 4: Apply the Inclusion-Exclusion Formula

\[
|A \cup B \cup C| = 621 + 414 + 248 - 206 - 125 - 82 + 41 = 911
\]

### Step 5: Subtract to Find the Complement

The total count of numbers not divisible by 2, 3, or 5 is:
\[
\text{Total not divisible} = 1241 - 911 = 330
\]

Therefore, there are **330 numbers** between 50 and 1290 that are not divisible by 2, 3, or 5.

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Do you want more details or have any questions?

Here are 5 related questions:
1. How would the calculation change if we were looking for numbers divisible by 2 or 5 only?
2. Can you explain how the principle of inclusion-exclusion applies in general?
3. What if we wanted to find numbers not divisible by 3 or 7?
4. How can we generalize this to find numbers not divisible by multiple sets of primes?
5. How can we use arithmetic sequences to simplify other similar problems?

**Tip:** When dealing with divisibility questions, consider the prime factors to simplify calculations.

How many numbers between 50 and 1290 are not divisible by 2 and 3 and 5?

Explore how to find the number of integers between 50 and 1290 that are not divisible by 2, 3, or 5 using the principle of inclusion and exclusion. This step-by-step solution covers concepts from number theory, arithmetic sequences, and provides detailed calculations for students in Grades 10-12.

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