find three every other even integers such that the sum of all three integers is equal to three times 

To find three every other even integers such that their sum is equal to three times something (though the exact multiplier is unclear), let's proceed with the following general approach:

### Step-by-step process:

1. **Define the integers**:
   Let the three every other even integers be:
   \[
   x, \quad x + 4, \quad x + 8
   \]
   where \(x\) is the first even integer, \(x + 4\) is the next even integer, and \(x + 8\) is the third even integer.

2. **Sum of the integers**:
   The sum of these three integers is:
   \[
   x + (x + 4) + (x + 8) = 3x + 12
   \]

3. **Relating to three times a number**:
   If the sum is supposed to be three times a certain number, say \(k\), then we set up the equation:
   \[
   3x + 12 = 3k
   \]

4. **Solve for \(x\)**:
   To find \(x\), divide both sides of the equation by 3:
   \[
   x + 4 = k
   \]
   \[
   x = k - 4
   \]

5. **Conclusion**:
   The three every other even integers are:
   \[
   (k - 4), \quad (k), \quad (k + 4)
   \]
   So, the solution depends on what \(k\) is.

### Missing information:
If you know the value of \(k\), I can calculate the integers for you.

Would you like to clarify the value for \(k\), or do you have any questions?

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**Follow-up questions:**
1. What is the value of \(k\) in this case?
2. Would you like to solve this problem with specific values of \(k\)?
3. Do you need help with another type of integer progression?
4. Are you interested in exploring odd integers instead?
5. Would you like to see the step-by-step solution for similar problems?

**Tip**: When dealing with arithmetic progressions, try to generalize the relation between consecutive terms to simplify your calculations.

Find three every other even integers such that the sum of all three integers is equal to three times something.

This page explains how to find three every other even integers whose sum equals three times a number. It includes step-by-step solutions for defining integers, solving a linear equation, and applying arithmetic progressions. Suitable for Grades 6-8.

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