Let the two-digit number be represented as $10t + u$, where $t$ is the tens digit and $u$ is the units digit.

Step 1: Setting up the first condition

The problem states that "the units digit exceeds the tens digit by 2." Therefore, we have the equation:

$u = t + 2$

Step 2: Setting up the second condition

The problem also states that "the number is 4 times the sum of its digits." The sum of the digits is $t + u$, and the number itself is $10t + u$. Therefore, the second equation is:

$10t + u = 4(t + u)$

Step 3: Substituting the first equation into the second

Substitute $u = t + 2$ into the second equation:

$10t + (t + 2) = 4(t + (t + 2))$

Simplify both sides:

$11t + 2 = 4(2t + 2)$

$11t + 2 = 8t + 8$

Step 4: Solving for $t$

Now, solve for $t$ by isolating it on one side of the equation:

$11t - 8t = 8 - 2$

$3t = 6$

$t = 2$

Step 5: Finding $u$

Now that we know $t = 2$, use the first equation to find $u$:

$u = t + 2 = 2 + 2 = 4$

Step 6: Verifying the solution

The number is $10t + u = 10(2) + 4 = 24$.

The sum of the digits is $2 + 4 = 6$, and four times the sum of the digits is $4 \times 6 = 24$, which matches the number itself.

Thus, the number is $\boxed{24}$.

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Would you like further details or have any questions?

Here are 5 related questions you might explore:

1. What if the units digit exceeded the tens digit by 3 instead of 2?
2. How can this approach be generalized to solve similar problems with different relationships between the digits?
3. What if the number was 5 times the sum of its digits instead of 4 times?
4. How can we solve this type of problem when given a three-digit number?
5. How does this problem relate to linear equations and algebraic techniques?

Tip: When solving word problems, clearly translate the conditions into equations to avoid confusion and simplify the solution process.