2. 外语
  3. If m and n are positive integers, is n even? (1) m(m + 2) + 1 = mn (2) m(m + n) is odd.

If m and n are positive integers, is n even? (1) m(m + 2) + 1 = mn (2) m(m + n) is odd.

admin2022-10-18  15
问题 If m and n are positive integers, is n even?
(1) m(m + 2) + 1 = mn
(2) m(m + n) is odd.
选项 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.
答案D
解析 (1)    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.
(2)    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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