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.