Let the integer be denoted by $x$. The next consecutive even integer would then be $x + 2$.

According to the problem:

• 8 times the next consecutive even integer is $8(x + 2)$.
• The difference between this value and the integer $x$ is given to be 72.

We can set up the equation:

$8(x + 2) - x = 72$

Now, solve the equation:

$8x + 16 - x = 72$ $7x + 16 = 72$ $7x = 72 - 16$ $7x = 56$ $x = 8$

So, the smaller integer is $x = 8$. The next consecutive even integer is:

$x + 2 = 8 + 2 = 10$

Thus, the greater integer is $\boxed{10}$.

Would you like further explanation, or do you have any other questions?

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Here are 5 related questions:

1. What would the result be if the difference was 100 instead of 72?
2. How would the equation change if we dealt with consecutive odd integers?
3. What if the given integer was subtracted from 6 times the next consecutive even integer instead?
4. How would you set up the equation if the difference was negative?
5. How does this problem change if the integer is negative?

Tip: When solving word problems, always define your variables clearly and create an equation based on the given conditions.