题目信息
If m and n are positive integers, is n even?
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.
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已选答案：
正确答案：
D:EACH statement ALONE is sufficient.
Arithmetic Properties of integers
Given that m(m + 2) + 1 = mn, then m cannot be even, since if m were even, then we would have an odd integer, namely m(m + 2) + 1, equal to an even integer, namely mn. Therefore, m is odd. Hence, m(m + 2) is odd, being the product of two odd integers, and thus m(m + 2) + 1 is even. Since m(m + 2) + 1 = mn, it follows that mn is even, and since m is odd, it follows that n is even; SUFFICIENT.
Alternatively, the table below shows that m(m + 2) + 1 = mn is only possible when m is odd and n is even.
Since m(m + n) is odd, it follows that m is odd and m + n is odd. Therefore, n = (m + n) − m is a difference of two odd integers and hence n is even; SUFFICIENT.
The correct answer is D;each statement alone is sufficient.
Given that m(m + 2) + 1 = mn, then m cannot be even, since if m were even, then we would have an odd integer, namely m(m + 2) + 1, equal to an even integer, namely mn. Therefore, m is odd. Hence, m(m + 2) is odd, being the product of two odd integers, and thus m(m + 2) + 1 is even. Since m(m + 2) + 1 = mn, it follows that mn is even, and since m is odd, it follows that n is even; SUFFICIENT.
Alternatively, the table below shows that m(m + 2) + 1 = mn is only possible when m is odd and n is even.
| m | n | m(m + 2) + 1 | mn |
| even | even | odd | even |
| even | odd | odd | even |
| odd | even | even | even |
| odd | odd | even | odd |
Since m(m + n) is odd, it follows that m is odd and m + n is odd. Therefore, n = (m + n) − m is a difference of two odd integers and hence n is even; SUFFICIENT.
The correct answer is D;each statement alone is sufficient.
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