Question

If the units digit of integer n is greater than 2, what is the units digit of n?
(1) The units digit of n is the same as the units digit of n².
(2) The units digit of n is the same as the units digit of n³.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

For questions involving units digits, focus on the cyclicity of the last digit for powers of numbers. For example, the units digits of powers of 4 are (4, 6, 4, 6, ...), and for 5, they are always 5. This can save time compared to calculating the full power.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
We are given an integer \(n\) whose units digit is greater than 2. This means the units digit can be 3, 4, 5, 6, 7, 8, or 9. We need to find the specific units digit of \(n\). Let U(\(x\)) denote the units digit of \(x\).
Step 2: Detailed Explanation:
Analyze Statement (1): The units digit of \(n\) is the same as the units digit of \(n^2\). This means U(\(n\)) = U(\(n^2\)). Let's test the possible units digits:

If U(\(n\)) = 3, U(\(n^2\)) = U(9) = 9. (No match)
If U(\(n\)) = 4, U(\(n^2\)) = U(16) = 6. (No match)
If U(\(n\)) = 5, U(\(n^2\)) = U(25) = 5. (Match)
If U(\(n\)) = 6, U(\(n^2\)) = U(36) = 6. (Match)
If U(\(n\)) = 7, U(\(n^2\)) = U(49) = 9. (No match)
If U(\(n\)) = 8, U(\(n^2\)) = U(64) = 4. (No match)
If U(\(n\)) = 9, U(\(n^2\)) = U(81) = 1. (No match)
From statement (1), the units digit of \(n\) could be 5 or 6. Since there are two possibilities, this statement is not sufficient.
Analyze Statement (2): The units digit of \(n\) is the same as the units digit of \(n^3\). This means U(\(n\)) = U(\(n^3\)). Let's test the possible units digits:

If U(\(n\)) = 3, U(\(n^3\)) = U(27) = 7. (No match)
If U(\(n\)) = 4, U(\(n^3\)) = U(64) = 4. (Match)
If U(\(n\)) = 5, U(\(n^3\)) = U(125) = 5. (Match)
If U(\(n\)) = 6, U(\(n^3\)) = U(216) = 6. (Match)
If U(\(n\)) = 7, U(\(n^3\)) = U(343) = 3. (No match)
If U(\(n\)) = 8, U(\(n^3\)) = U(512) = 2. (No match, and also 2 is not>2)
If U(\(n\)) = 9, U(\(n^3\)) = U(729) = 9. (Match)
From statement (2), the units digit of \(n\) could be 4, 5, 6, or 9. Since there are multiple possibilities, this statement is not sufficient.
Analyze Both Statements Together:
From statement (1), the possible units digits are {5, 6}.
From statement (2), the possible units digits are {4, 5, 6, 9}.
The common possibilities that satisfy both statements are {5, 6}. Since there are still two possible values for the units digit of \(n\), the statements together are not sufficient.
Step 3: Final Answer:
Since combining both statements still does not yield a unique units digit for \(n\), the information is insufficient.