how many integers are multiples of 7 from 60 to 290

To find how many integers are multiples of 7 between 60 and 290, follow these steps:

1. **Find the smallest multiple of 7 greater than or equal to 60:**
   - Divide 60 by 7: \( \frac{60}{7} \approx 8.57 \)
   - The smallest integer greater than 8.57 is 9.
   - The smallest multiple of 7 is \( 7 \times 9 = 63 \).

2. **Find the largest multiple of 7 less than or equal to 290:**
   - Divide 290 by 7: \( \frac{290}{7} \approx 41.43 \)
   - The largest integer less than 41.43 is 41.
   - The largest multiple of 7 is \( 7 \times 41 = 287 \).

3. **Count the number of multiples of 7:**
   - The sequence of multiples of 7 between 60 and 290 is: 63, 70, 77, ..., 287.
   - These are multiples of 7 from \( 7 \times 9 \) to \( 7 \times 41 \).
   - To find the number of terms in this sequence, use the formula for the number of terms in an arithmetic sequence: \( n = \text{last term index} - \text{first term index} + 1 \).
   - \( n = 41 - 9 + 1 = 33 \).

Thus, there are **33 integers** that are multiples of 7 between 60 and 290.

Would you like more details, or have any questions?

Here are some related questions you might find interesting:
1. What is the sum of all multiples of 7 between 60 and 290?
2. How many multiples of 7 are there between 100 and 500?
3. What is the smallest multiple of 7 greater than 1000?
4. How many integers between 1 and 1000 are multiples of both 7 and 5?
5. What is the largest prime factor of 287?

**Tip:** When finding multiples of a number within a range, always remember to round up for the smallest multiple and round down for the largest multiple.

Learn how to count the number of integers that are multiples of 7 between 60 and 290 with a detailed step-by-step guide. This problem involves arithmetic sequences and finding multiples within a given range. Suitable for students in Grades 7-9.

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