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.