Question

If “m” is a positive integer, is \( m^2 + 1 \) when divided by 10 leaves remainder ZERO?
(1) \( 101^{16} \times m \), when divided by 2 leaves a remainder 1.
(2) \( 101^{16} \times m \), when divided by 5 leaves a remainder 2.

• 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 but neither statement is sufficient alone.
• D. Each statement alone is sufficient to answer the question.
• E. Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

When dealing with remainders and divisibility, use the system of congruences to combine multiple conditions.

Answer 1

The Correct Option is C

Step 1: Analyzing Statement 1.
Statement 1 tells us that \( 101^{16} \times m \) when divided by 2 leaves a remainder of 1. However, this statement does not give us enough information about \( m^2 + 1 \) when divided by 10. So, statement 1 alone is insufficient.
Step 2: Analyzing Statement 2.
Statement 2 tells us that \( 101^{16} \times m \) when divided by 5 leaves a remainder of 2. This statement does not directly address the divisibility of \( m^2 + 1 \) by 10 either. Therefore, statement 2 alone is also insufficient.
Step 3: Combining Both Statements.
By combining both statements, we can form a system of congruences involving \( m \), and solve for \( m \) in relation to 10. Thus, the combined statements provide sufficient information to answer the question.
Step 4: Conclusion.
The correct answer is (C).