What is the smallest digit that can replace the symbol * in the number 63576*2 so that the resulting number is divisible by 8?

Introduction / Context:
This problem checks the rule of divisibility by 8, specifically how only the last three digits of a number determine whether the entire number is divisible by 8. The question embeds an unknown digit represented by * and asks for the smallest digit that makes the number divisible by 8.

Given Data / Assumptions:
- The number is 63576*2, where * represents a single digit from 0 to 9.
- We want the resulting full number to be divisible by 8.
- We are asked for the smallest possible digit that satisfies this condition.

Concept / Approach:
A number is divisible by 8 if and only if the integer formed by its last three digits is divisible by 8. Therefore, we only need to examine the last three digits of 63576*2, which are 6, *, and 2. We will treat these as a three digit number 6*2 and search for the smallest digit that makes this three digit number divisible by 8.

Step-by-Step Solution:
Let the digit * be x, so the last three digits form the number 6x2. This can be written as 600 + 10x + 2, which is 602 + 10x. We need 602 + 10x to be divisible by 8. Test x from 0 to 9. When x = 3, we get 602 + 30 = 632, and 632 / 8 = 79 exactly. Therefore, 6 3 2 is divisible by 8, so the whole number 6357632 is divisible by 8.

Verification / Alternative Check:
We can also test x = 7, giving 602 + 70 = 672, which is also divisible by 8, but 7 is larger than 3. Since the question asks for the smallest possible digit, 3 is preferred. A quick long division of 632 by 8 confirms the divisibility without remainder.

Why Other Options Are Wrong:
If x = 1, the last three digits form 612, and 612 / 8 leaves a remainder.
If x = 2, the last three digits form 622, which is not divisible by 8.
If x = 4, the last three digits form 642, which also is not divisible by 8.
Digit 0 gives 602, which is not divisible by 8 either. Therefore, only x = 3 works and is the smallest such digit.

Common Pitfalls:
Students often try to test the entire number instead of focusing only on the last three digits, which wastes time. Another common mistake is to misapply the divisibility rule by using the last digit or last two digits instead of the last three for 8. Some learners also forget to check all smaller digits once one working digit is found, which can lead to missing the smallest valid option.

Final Answer:
The smallest digit that can replace * so that 63576*2 is divisible by 8 is 3.