To find how many integers between 35 and 500 are divisible by 7, follow these steps:

1. Identify the smallest multiple of 7 greater than or equal to 35:

   $\text{Smallest multiple of 7} \geq 35: \quad \frac{35}{7} = 5 \quad \text{(exact, so 35 is a multiple of 7)}$

   Thus, the smallest multiple is 35.

2. Identify the largest multiple of 7 less than or equal to 500:

   $\text{Largest multiple of 7} \leq 500: \quad \frac{500}{7} \approx 71.4286$

   So, the largest multiple is $7 \times 71 = 497$.

3. Calculate the number of multiples of 7 between 35 and 497:

   The sequence of numbers divisible by 7 between 35 and 497 is an arithmetic sequence with the first term $a_1 = 35$ and the common difference $d = 7$.

   The $n$-th term of the sequence can be represented by: $a_n = a_1 + (n-1) \times d$ Setting $a_n = 497$, we get: $497 = 35 + (n-1) \times 7$ Solving for $n$: $497 - 35 = (n-1) \times 7$ $462 = (n-1) \times 7$ $n-1 = \frac{462}{7} = 66$ $n = 67$

Thus, there are 67 integers between 35 and 500 that are divisible by 7.

Would you like any more details or explanations?

Here are five related questions you might find interesting:

1. How many integers between 50 and 1000 are divisible by 5?
2. What is the sum of all integers between 1 and 100 that are divisible by 3?
3. How many even integers are there between 200 and 600?
4. What is the smallest positive integer divisible by both 7 and 9?
5. How many integers between 1 and 1000 are divisible by both 3 and 5?

Tip: To quickly determine how many numbers in a range are divisible by a specific number, consider the smallest and largest multiples of the divisor within the range and calculate how many terms exist in that sequence.