how many integers are there between 35 and 500 which are divisible by 7

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.

Discover how many integers between 35 and 500 are divisible by 7 with a step-by-step solution. Learn about arithmetic sequences and divisibility rules. Suitable for students in Grades 6-8.

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