Let's denote the three consecutive even integers as $x$, $x + 2$, and $x + 4$. The sum of these integers is given as 174. Therefore, we can write the equation:

$x + (x + 2) + (x + 4) = 174$

Now, simplify and solve for $x$:

$x + x + 2 + x + 4 = 174$

$3x + 6 = 174$

Subtract 6 from both sides:

$3x = 168$

Now, divide by 3:

$x = 56$

So, the first number is 56.

Would you like details or have any questions?

Here are 8 related questions:

1. What are the other two consecutive even integers?
2. What is the sum of the first two consecutive even integers?
3. If the sum was 200, what would the first number be?
4. What if the integers were consecutive odd integers instead?
5. How do you find consecutive even integers given a total sum?
6. What happens if you increase the first number by 4?
7. How does this method change for consecutive multiples of 3?
8. Can you form an equation for four consecutive even integers?

Tip: For any sequence of consecutive even (or odd) numbers, you can always express them in terms of a single variable like this.