Which of the following digits cannot appear in the units place of a perfect square?

Introduction / Context:
This is a conceptual question about the possible last digits of perfect squares. It asks which digit cannot appear in the units place of a perfect square. Knowing the pattern of units digits for squares of integers from 0 to 9 allows us to answer quickly.

Given Data / Assumptions:

- We are dealing with whole numbers whose squares are perfect squares.

- We only care about the units digit of the square.

- Options are the digits 4, 6, 8, and 9.

Concept / Approach:
Any integer can be written in terms of its last digit from 0 to 9. We can square each of these digits and observe the units digit of the result. This reveals the complete set of possible units digits of any perfect square. Any digit not in this set cannot be the units digit of a perfect square.

Step-by-Step Solution:
Step 1: List digits 0 to 9 and compute their squares. 0^2 = 0 → units digit 0. 1^2 = 1 → units digit 1. 2^2 = 4 → units digit 4. 3^2 = 9 → units digit 9. 4^2 = 16 → units digit 6. 5^2 = 25 → units digit 5. 6^2 = 36 → units digit 6. 7^2 = 49 → units digit 9. 8^2 = 64 → units digit 4. 9^2 = 81 → units digit 1. Step 2: Collect possible units digits of any perfect square. These are 0, 1, 4, 5, 6, 9. Step 3: Compare with the options. Among 4, 6, 8, 9, we see that 8 does not appear in the list above.

Verification / Alternative check:
Any perfect square is the square of some integer ending with a digit 0 to 9. Since none of these give a units digit of 8 when squared, no perfect square can end with 8. This confirms that 8 is impossible as the units digit of a perfect square.

Why Other Options Are Wrong:
- 4: Appears from 2^2 and 8^2, so it is a valid units digit of a perfect square.
- 6: Appears from 4^2 and 6^2, so it is also valid.

- 9: Appears from 3^2 and 7^2, so it is valid as well.

Common Pitfalls:
Some learners try to reason with a few examples and may miss some digits. Others wrongly assume that all digits can occur as units digits of squares. It is safer to systematically compute squares of 0 to 9, which fully covers all possibilities due to how place value works.

Final Answer:
The digit that cannot appear in the units place of a perfect square is 8.