find the sum of all positive integers less than 600 which are divisible by 8

To find the sum of all positive integers less than 600 that are divisible by 8, follow these steps:

1. **Identify the smallest and largest multiples of 8:**

   - The smallest multiple of 8 is \(8 \times 1 = 8\).
   - The largest multiple of 8 less than 600 can be calculated by dividing 600 by 8 and taking the floor of the result:
     \[
     \frac{600}{8} = 75 \quad \text{(since \(75 \times 8 = 600\) is exactly 600, but we want less than 600, so consider 74 instead)}
     \]
     Therefore, the largest multiple is \(8 \times 74 = 592\).

2. **Determine the sequence of numbers divisible by 8:**

   The sequence of numbers divisible by 8 between 8 and 592 is an arithmetic sequence where:
   - The first term \(a_1 = 8\),
   - The last term \(a_n = 592\),
   - The common difference \(d = 8\).

3. **Find the number of terms in the sequence:**

   The \( n \)-th term of the sequence can be expressed as:
   \[
   a_n = a_1 + (n-1) \times d
   \]
   Setting \( a_n = 592 \):
   \[
   592 = 8 + (n-1) \times 8
   \]
   Solving for \( n \):
   \[
   592 - 8 = (n-1) \times 8
   \]
   \[
   584 = (n-1) \times 8
   \]
   \[
   n-1 = \frac{584}{8} = 73
   \]
   \[
   n = 74
   \]

   So, there are 74 terms in this sequence.

4. **Calculate the sum of the sequence:**

   The sum \( S_n \) of the first \( n \) terms of an arithmetic sequence can be found using the formula:
   \[
   S_n = \frac{n}{2} \times (a_1 + a_n)
   \]
   Substituting the values:
   \[
   S_{74} = \frac{74}{2} \times (8 + 592) = 37 \times 600 = 22200
   \]

Thus, the sum of all positive integers less than 600 that are divisible by 8 is **22,200**.

Would you like any more details or explanations?

Here are five related questions you might find interesting:
1. Find the sum of all positive integers less than 1000 that are divisible by 9.
2. How many integers between 50 and 200 are divisible by 6?
3. What is the sum of all even integers less than 1000?
4. Find the sum of all multiples of 5 between 100 and 500.
5. What is the sum of all positive integers less than 500 divisible by 12?

**Tip:** The sum of an arithmetic sequence can be quickly calculated if you know the first term, the last term, and the number of terms. This is a common method for summing sequences of numbers divisible by a particular integer.

Learn how to find the sum of all positive integers less than 600 that are divisible by 8. Follow a step-by-step solution using arithmetic sequences and the summation formula. Suitable for students in Grades 7-9.

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