Question

If 'm' is a positive integer, is "m\(^2\) + 1" divisible by 10 (leaves remainder ZERO)?
I. When m is divided by 2, it leaves a remainder of 1.
II. When m is divided by 5, it leaves a remainder of 2.

• A. Statement I alone is sufficient but statement II alone is not sufficient to answer the question asked.
• B. Statement II alone is sufficient but statement I alone is not sufficient to answer the question asked.
• C. Both statements I and II 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 I and II are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

Questions about divisibility by 10 are almost always questions about the units digit. Rephrasing the question in terms of the units digit of the variable can make the problem much clearer.

Answer 1

The Correct Option is C

Note: The provided text for the statements in the original document is heavily corrupted by OCR errors. The solution below is based on a logical reconstruction that aligns with the structure of such problems.
Step 1: Understanding the Concept:
This is a "Yes/No" Data Sufficiency question about divisibility and remainders. The question asks if \(m^2 + 1\) is divisible by 10. For a number to be divisible by 10, its units digit must be 0. This means that \(m^2 + 1\) must end in 0, which in turn means that \(m^2\) must end in 9. For an integer's square (\(m^2\)) to end in 9, the integer (\(m\)) itself must have a units digit of 3 or 7. So, the question is equivalent to: "Does the integer m end in 3 or 7?"
Step 2: Detailed Explanation:
Analyze Reconstructed Statement I: "When m is divided by 2, it leaves a remainder of 1."
This means that \(m\) is an odd number. The units digit of an odd number can be 1, 3, 5, 7, or 9.

If m ends in 3 or 7, the answer is "Yes".
If m ends in 1, 5, or 9, the answer is "No". (e.g., if m=1, \(m^2+1=2\); if m=5, \(m^2+1=26\))
Since we cannot determine a definite yes or no, Statement I is not sufficient.
Analyze Reconstructed Statement II: "When m is divided by 5, it leaves a remainder of 2."
This means that the units digit of \(m\) must be either 2 or 7.

If m ends in 7, the answer is "Yes". (e.g., if m=7, \(m^2+1=50\), which is divisible by 10)
If m ends in 2, the answer is "No". (e.g., if m=2, \(m^2+1=5\), which is not divisible by 10)
Since we cannot determine a definite yes or no, Statement II is not sufficient.
Analyze Statements I and II Together:
From Statement I, we know \(m\) is odd. From Statement II, we know \(m\) ends in 2 or 7. Combining these two conditions, the only possibility is that the units digit of \(m\) must be 7. If \(m\) has a units digit of 7, then \(m^2\) will have a units digit of 9, and \(m^2 + 1\) will have a units digit of 0. Therefore, \(m^2 + 1\) will be divisible by 10. This gives a definite "Yes" to the question. Therefore, both statements together are sufficient.
Step 3: Final Answer:
Neither statement alone is sufficient, but together they are sufficient. This corresponds to option (C).