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 \( n^2 \).
(2) The units digit of \( n \) is the same as the units digit of \( n^3 \).

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are not sufficient

Hint:

When solving problems involving digits, consider examining the units digits of powers of numbers to help eliminate possibilities.

Answer 1

The Correct Option is C

Step 1: Analyzing statement (1).
Statement (1) tells us that the units digit of \( n \) is the same as the units digit of \( n^2 \).
Let’s check the possible units digits of \( n \) and \( n^2 \):
- If \( n \) ends in 0, \( n^2 \) ends in 0.
- If \( n \) ends in 1, \( n^2 \) ends in 1.
- If \( n \) ends in 2, \( n^2 \) ends in 4.
- If \( n \) ends in 3, \( n^2 \) ends in 9.
- If \( n \) ends in 4, \( n^2 \) ends in 6.
- If \( n \) ends in 5, \( n^2 \) ends in 5.
- If \( n \) ends in 6, \( n^2 \) ends in 6.
- If \( n \) ends in 7, \( n^2 \) ends in 9.
- If \( n \) ends in 8, \( n^2 \) ends in 4.
- If \( n \) ends in 9, \( n^2 \) ends in 1.
Therefore, if the units digit of \( n \) is the same as the units digit of \( n^2 \), then \( n \) must end in 1, 5, or 6. This does not uniquely determine \( n \) because there are multiple possibilities.
Step 2: Analyzing statement (2).
Statement (2) tells us that the units digit of \( n \) is the same as the units digit of \( n^3 \).
Let’s check the possible units digits of \( n \) and \( n^3 \):
- If \( n \) ends in 0, \( n^3 \) ends in 0.
- If \( n \) ends in 1, \( n^3 \) ends in 1.
- If \( n \) ends in 2, \( n^3 \) ends in 8.
- If \( n \) ends in 3, \( n^3 \) ends in 7.
- If \( n \) ends in 4, \( n^3 \) ends in 4.
- If \( n \) ends in 5, \( n^3 \) ends in 5.
- If \( n \) ends in 6, \( n^3 \) ends in 6.
- If \( n \) ends in 7, \( n^3 \) ends in 3.
- If \( n \) ends in 8, \( n^3 \) ends in 2.
- If \( n \) ends in 9, \( n^3 \) ends in 9.
Therefore, if the units digit of \( n \) is the same as the units digit of \( n^3 \), then \( n \) must end in 0, 1, 5, or 6. This does not uniquely determine \( n \) because there are multiple possibilities.
Step 3: Combining statements (1) and (2).
From statement (1), \( n \) must end in 1, 5, or 6.
From statement (2), \( n \) must end in 0, 1, 5, or 6.
Thus, combining both statements, the units digit of \( n \) must be 1, 5, or 6. \[ \boxed{C} \]